Key Concept: Linear Equations in One Variable, Algebraic Expressions

a) -3

[Solution Description]

Find a common denominator for the fractions, which is 20. Multiply each term by 20 to eliminate the denominators:

$20 \cdot \left( \frac{y + 3}{4} \right) - 20 \cdot \left( \frac{y - 2}{5} \right) = 20 \cdot 1$

Simplify each term:

$5(y + 3) - 4(y - 2) = 20$

Distribute the constants:

$5y + 15 - 4y + 8 = 20$

Combine like terms:

$y + 23 = 20$

Subtract 23 from both sides:

$y = -3$

Therefore, the solution is $y = -3$.

Key Concept: Linear Expressions

b) $4y - 7$

[Solution Description] A linear expression is an expression where the highest power of the variable is 1.

Key Concept: Linear Equations in One Variable

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The equation $2x - 3 = 7$ is a linear equation in one variable because it contains only one variable, $x$, and the highest power of $x$ is 1. This satisfies the definition of a linear equation in one variable. Therefore, both the Assertion and the Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Linear Equations in One Variable, Algebraic Expressions

a) 7

[Solution Description]

Start by expanding both sides of the equation:

$6x - 10 = 4x + 12 - 8$

Simplify the right side:

$6x - 10 = 4x + 4$

Subtract $4x$ from both sides:

$2x - 10 = 4$

Add 10 to both sides:

$2x = 14$

Divide both sides by 2:

$x = 7$

Therefore, the solution is $x = 7$.

Key Concept: Linear Equations in One Variable

b) $\frac{y}{2} + 5 = 10$

[Solution Description] A linear equation in one variable is an equation that can be written in the form $ax + b = 0$, where $a$ and $b$ are constants, and $x$ is the variable. The highest power of the variable should be 1.

Key Concept: Left Hand Side and Right Hand Side

a) $\frac{y}{2} + \frac{z}{2}$

[Solution Description]

In an equation, the Left Hand Side (LHS) is the expression on the left of the equality (\$) sign. For the equation $\frac{y}{2} + \frac{z}{2} = 6$, the LHS is $\frac{y}{2} + \frac{z}{2}$. The other options are either the RHS or incorrect combinations.

Key Concept: Linear Equations in One Variable

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that $3x - 5 = 10$ is a linear equation in one variable. This is true because it fits the definition of a linear equation, which involves an equality sign and has only one variable with the highest power of 1. The reason provided is also correct, as a linear equation in one variable indeed has the highest power of the variable as 1. Furthermore, the reason correctly explains why the assertion is true. Therefore, both the assertion and the reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Linear Equation in One Variable

a) $4x - 7 = 9$

[Solution Description]

A linear equation in one variable is an equation where the highest power of the variable is 1 and there is only one variable. From the given options, $4x - 7 = 9$ is a linear equation in one variable ($x$) because the highest power of $x$ is 1. The other options either have more than one variable or are not equations.

Key Concept: Definition of algebraic expressions and equations, Linear equations in one variable

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that $3x + 5 = 2x - 7$ is a linear equation in one variable. This is true because the equation involves only one variable $x$ and the highest power of $x$ is 1, which satisfies the definition of a linear equation in one variable. The reason given is that all equations involving only one variable with the highest power of 1 are considered linear equations in one variable. This is also true and correctly explains why the assertion is valid.

Key Concept: Linear Equations in One Variable, Algebraic Expressions

b) $10$

[Solution Description]

To solve for $z$, first multiply both sides by $2$:

$\frac{z - 4}{2} \times 2 = 3 \times 2$

Simplifying gives:

$z - 4 = 6$

Next, add $4$ to both sides:

$z - 4 + 4 = 6 + 4$

This results in:

$z = 10$

Therefore, the solution is $z = 10$.

Key Concept: Difference between expressions and equations

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

The assertion states that an equation always has an equality sign, which is true because an equation is defined as a statement that asserts the equality of two expressions. The reason states that an expression can have variables but does not include an equality sign, which is also true because an expression is a combination of numbers, variables, and operators without any equality. However, the reason does not explain why an equation always has an equality sign; it simply describes a characteristic of expressions. Therefore, both the assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Expressions vs Equations, Linear Equations

d) An equation must include an equality sign, while an expression does not.

[Solution Description]

An expression is a combination of numbers, variables, and operations without an equality sign, while an equation includes an equality sign and sets two expressions equal to each other. Here, $3x + 5$ is an expression because it does not have an equality sign. $3x + 5 = 20$ is an equation because it has an equality sign and sets $3x + 5$ equal to $20$.

Key Concept: Difference between expressions and equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the expression $3x + 5$ can be converted into an equation by adding an equality sign and a constant term on the right-hand side. This is true because, for example, $3x + 5 = 10$ is a valid equation derived from the expression $3x + 5$. The reason given is that an equation is formed when an expression is equated to another expression or a constant. This is also true as per the definition of an equation. Moreover, the reason correctly explains the assertion because it provides the general rule that allows the conversion of an expression into an equation by introducing an equality sign and another expression or constant. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Linear expressions

a) $5x + 3$

[Solution Description] A linear expression in one variable has the highest power of the variable equal to 1. Option (a) is a linear expression because the highest power of \$x\$ is 1. Options (b), (c), and (d) are not linear because they either have more than one variable or the highest power of the variable is greater than 1.

Key Concept: Assertion-Reason

d) Assertion is false, but Reason is true.

[Solution Description]

The assertion states that the expression $3x + 2y$ is a linear equation in one variable. However, this is incorrect because $3x + 2y$ contains two variables, $x$ and $y$. A linear equation in one variable must have only one variable with its highest power as 1. The reason states that a linear equation in one variable has the highest power of the variable as 1, which is correct. Therefore, the assertion is false, but the reason is true.

Key Concept: Variables and Constants

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that $3x + 5$ is a linear expression in one variable. This is true because it only contains one variable, $x$, and the highest power of $x$ is 1. The reason given is that a linear expression in one variable has the highest power of the variable equal to 1. This is also true and correctly explains why $3x + 5$ is a linear expression in one variable.

Key Concept: Linear Equations

b) $x = 5$

[Solution Description]

To solve the equation $3x - 5 = 10$, follow these steps:

Step 1: Add 5 to both sides of the equation to isolate the term with x.

$3x - 5 + 5 = 10 + 5$

$3x = 15$

Step 2: Divide both sides by 3 to solve for x.

$\frac{3x}{3} = \frac{15}{3}$

$x = 5$

Key Concept: Linear equations, variables

c) 6

[Solution Description]

Let the unknown number be $x$. According to the problem, twice the number plus 5 equals 17. This can be written as:

$2x + 5 = 17$

Subtract 5 from both sides to isolate the term with $x$:

$2x = 12$

Divide both sides by 2 to solve for $x$:

$x = 6$

Therefore, the number is 6.

Key Concept: Linear Equations in One Variable

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that $3x - 5 = 10$ is a linear equation in one variable. This is true because the equation contains only one variable, x, and its highest power is 1. The reason provided is also correct, as a linear equation in one variable must have the highest power of the variable equal to 1. Moreover, the reason correctly explains why the assertion is true, as the definition of a linear equation in one variable directly supports the assertion.

Key Concept: Equations, solving for variables

c) 20

[Solution Description]

Start with the equation:

$\frac{y}{2} - 3 = 7$

Add 3 to both sides to isolate the term with $y$:

$\frac{y}{2} = 10$

Multiply both sides by 2 to solve for $y$:

$y = 20$

Therefore, the solution is $y = 20$.

Key Concept: Equality Sign

b) Presence of an equality sign

[Solution Description]

An equation is defined as a mathematical statement that includes an equality (\$) sign. The equality sign separates the Left Hand Side (LHS) and the Right Hand Side (RHS) of the equation. Let us check each option:

- Presence of variables: This is not the defining feature because expressions can also have variables.

- Equality sign: This is the defining feature of an equation.

- Highest power of variable: This is not the defining feature because equations can be of any degree.

- Use of letters: This is not the defining feature because letters can be used in both equations and expressions.

Therefore, the correct answer is the presence of an equality sign.

Key Concept: Linear Expressions

a) $4x + 3$

[Solution Description]

A linear expression is an expression where the highest power of the variable is 1. Let us check each option:

- $4x + 3$ is a linear expression because the highest power of $x$ is 1.

- $x^2 + 5$ is not a linear expression because the highest power of $x$ is 2.

- $2y^2 + y$ is not a linear expression because the highest power of $y$ is 2.

- $3z + z^2$ is not a linear expression because the highest power of $z$ is 2.

Therefore, the correct answer is $4x + 3$.

Key Concept: Linear Equations, Algebraic Expressions

b) $2x - 3 = 7$

[Solution Description]

A linear equation in one variable has the highest power of the variable as 1 and contains only one variable. Let's analyze each option:

1. $x + y = 10$: This equation involves two variables, so it is not linear in one variable.

2. $2x - 3 = 7$: The highest power of $x$ is 1 and it contains only one variable, so it is linear.

3. $x^2 - 4 = 0$: The highest power of $x$ is 2, so it is not linear.

4. $y^2 + y = 6$: The highest power of $y$ is 2, so it is not linear.

Therefore, the correct answer is $2x - 3 = 7$.

Key Concept: Linear Equations in One Variable

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the equation $3x + 5 = 20$ is a linear equation in one variable. This is true because it contains only one variable $x$ and its highest power is 1. The reason explains that a linear equation in one variable has the highest power of the variable equal to 1, which is also true. Furthermore, the reason correctly explains why the assertion is true. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Linear Equations, Algebraic Expressions

b) $3y + 7 = 10$

[Solution Description]

A linear equation in one variable has the highest power of the variable as 1. Let's analyze each option:

1. $x^2 + 2x = 5$: The highest power of $x$ is 2, so it is not linear.

2. $3y + 7 = 10$: The highest power of $y$ is 1, so it is linear.

3. $z^3 - z = 0$: The highest power of $z$ is 3, so it is not linear.

4. $2xy = 8$: This equation involves two variables, so it is not linear in one variable.

Therefore, the correct answer is $3y + 7 = 10$.

Key Concept: Solving Equations Having Variable on Both Sides

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To solve the equation $3x + 4 = 2x - 1$, we first transpose the term $2x$ from the right-hand side (RHS) to the left-hand side (LHS) by subtracting $2x$ from both sides. This gives us:

$3x - 2x + 4 = -1$

Simplifying the left side, we get:

$x + 4 = -1$

Next, we isolate the variable $x$ by subtracting 4 from both sides:

$x = -1 - 4$

$x = -5$

Therefore, the equation has a unique solution, which is $x = -5$. This supports the assertion that the equation has a unique solution. The reason explains the method of isolating the variable by transposing terms, which is a valid and correct explanation for why the equation has a unique solution.

Key Concept: Variable on Both Sides, Fractional Equations

c) $x = \frac{29}{2}$

[Solution Description]

Start with the equation:

$\frac{4x - 7}{3} = \frac{2x + 5}{2}$

Multiply both sides by 6 (the LCM of 3 and 2) to eliminate denominators:

$6 \times \frac{4x - 7}{3} = 6 \times \frac{2x + 5}{2}$

Simplify to obtain:

$2(4x - 7) = 3(2x + 5)$

Expand both sides:

$8x - 14 = 6x + 15$

Subtract $6x$ from both sides:

$8x - 6x - 14 = 15$

Simplify to obtain:

$2x - 14 = 15$

Add $14$ to both sides:

$2x = 15 + 14$

Simplify to obtain:

$2x = 29$

Divide both sides by 2:

$x = \frac{29}{2}$

Key Concept: Solving Equations with Variables on Both Sides

b) $4$

[Solution Description]

Multiply both sides by $2$ to eliminate the denominators:

$2 \times \left(\frac{4x - 3}{2}\right) = 2 \times \left(\frac{3x + 1}{2}\right)$

Simplify to get:

$4x - 3 = 3x + 1$

Subtract $3x$ from both sides:

$4x - 3x - 3 = 3x - 3x + 1$

Simplify to get:

$x - 3 = 1$

Add $3$ to both sides:

$x - 3 + 3 = 1 + 3$

This simplifies to:

$x = 4$

Key Concept: Solving Equations with Variables on Both Sides

c) 7

[Solution Description]

To solve the equation $4x - 7 = x + 14$, we first subtract $x$ from both sides to get:

$4x - x - 7 = 14$

Simplifying, we have:

$3x - 7 = 14$

Next, add $7$ to both sides:

$3x = 14 + 7$

So,

$3x = 21$

Finally, divide both sides by $3$:

$x = \frac{21}{3} = 7$

The solution is:

$x = 7$

Key Concept: Solving Equations with Variables on Both Sides

c) $-9$

[Solution Description]

Start by subtracting $2x$ from both sides to collect like terms:

$3x - 2x + 4 = 2x - 2x - 5$

Simplify to get:

$x + 4 = -5$

Next, subtract $4$ from both sides to isolate $x$:

$x + 4 - 4 = -5 - 4$

This simplifies to:

$x = -9$

Key Concept: Balancing Equations

b) -5

[Solution Description]

To solve the equation $3x + 4 = 2x - 1$, we need to isolate the variable $x$.

Step 1: Subtract $2x$ from both sides to get:

$$3x - 2x + 4 = -1$$

$$x + 4 = -1$$

Step 2: Subtract $4$ from both sides to isolate $x$:

$$x = -1 - 4$$

$$x = -5$$

The solution to the equation is $x = -5$.

Key Concept: Balancing Equations

c) 6

[Solution Description]

To solve the equation $5x - 7 = 3x + 5$, we need to isolate the variable $x$.

Step 1: Subtract $3x$ from both sides:

$$5x - 3x - 7 = 5$$

$$2x - 7 = 5$$

Step 2: Add $7$ to both sides:

$$2x = 12$$

Step 3: Divide both sides by 2:

$$x = 6$$

The solution to the equation is $x = 6$.

Key Concept: Balancing Equations

b) $x = 5$

[Solution Description]

Start with the equation: $3x + 5 = 2x + 10$. Subtract $2x$ from both sides to get $3x - 2x + 5 = 10$, which simplifies to $x + 5 = 10$. Next, subtract $5$ from both sides to isolate $x$: $x = 10 - 5$, so $x = 5$.

Key Concept: Balancing Equations

a) $\frac{11}{8}$

[Solution Description]

To solve the equation $\frac{4x - 3}{2} = \frac{2x + 1}{3}$, we first eliminate the fractions by multiplying both sides by 6 (the least common multiple of 2 and 3):

Step 1: Multiply both sides by 6:

$$6 \times \left(\frac{4x - 3}{2}\right) = 6 \times \left(\frac{2x + 1}{3}\right)$$

$$3(4x - 3) = 2(2x + 1)$$

Step 2: Expand both sides:

$$12x - 9 = 4x + 2$$

Step 3: Subtract $4x$ from both sides:

$$12x - 4x - 9 = 2$$

$$8x - 9 = 2$$

Step 4: Add $9$ to both sides:

$$8x = 11$$

Step 5: Divide both sides by 8:

$$x = \frac{11}{8}$$

The solution to the equation is $x = \frac{11}{8}$.

Key Concept: Balancing Equations

c) $x = 5$

[Solution Description]

Start with the equation: $5x - 4 = 3x + 6$. Subtract $3x$ from both sides: $5x - 3x - 4 = 6$, which simplifies to $2x - 4 = 6$. Add $4$ to both sides: $2x = 6 + 4$, so $2x = 10$. Divide both sides by $2$: $x = 5$.

Key Concept: Solving Equations with Variables on Both Sides, Transposing Terms

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

To solve the equation $3x + 4 = 2x - 5$, we need to isolate the variable $x$. Start by transposing $2x$ to the left-hand side (LHS) and $4$ to the right-hand side (RHS). This involves subtracting $2x$ from both sides and adding $5$ to both sides.

Step 1: Subtract $2x$ from both sides:

$3x - 2x + 4 = 2x - 2x - 5$

Simplifies to:

$x + 4 = -5$

Step 2: Add $5$ to both sides:

$x + 4 + 5 = -5 + 5$

Simplifies to:

$x + 9 = 0$

Step 3: Subtract $9$ from both sides:

$x + 9 - 9 = 0 - 9$

Simplifies to:

$x = -9$

The solution to the equation is $x = -9$. The assertion correctly states that the equation can be solved by transposing $2x$ to LHS and $4$ to RHS. The reason is also true, as transposing terms indeed involves moving terms from one side to the other while maintaining equality. However, the reason does not explain why transposing works in this specific case; it only defines what transposing is. Therefore, both the assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Transposing terms

b) $5$

[Solution Description]

To solve the equation $5x - 3 = 3x + 7$, we need to transpose the terms involving $x$ to one side and the constant terms to the other side.

Step 1: Subtract $3x$ from both sides of the equation:

$5x - 3x - 3 = 7$

This simplifies to:

$2x - 3 = 7$

Step 2: Add $3$ to both sides:

$2x = 7 + 3$

$2x = 10$

Step 3: Divide both sides by $2$:

$x = \frac{10}{2}$

$x = 5$

Thus, the solution to the equation is $x = 5$.

Key Concept: Solving Equations Having Variable on Both Sides

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To solve the equation $3x + 4 = 2x - 1$, we need to isolate the variable $x$.

Step 1: Transpose $2x$ to the left side by subtracting $2x$ from both sides:

$3x - 2x + 4 = -1$

Simplifying, we get:

$x + 4 = -1$

The assertion states that transposing $2x$ to the left side gives $x + 4 = -1$, which is correct as shown in the solution.

The reason given is that transposing a term involves moving it from one side of the equation to the other by changing its sign. This is also correct and explains why the assertion is true.

Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Solving Linear Equations

b) 8

[Solution Description]

To solve the equation $5x - 4 = 3x + 12$, we first subtract $3x$ from both sides:

$5x - 3x - 4 = 3x - 3x + 12$. This simplifies to:

$2x - 4 = 12$. Next, add 4 to both sides:

$2x = 12 + 4$. This gives:

$2x = 16$. Finally, divide both sides by 2 to find:

$x = \frac{16}{2}$. Therefore, the solution is:

$x = 8$.

Key Concept: Solving Equations with Variable on Both Sides

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let us solve the given equation step by step:

1. The original equation is $3x + 4 = 2x - 1$.

2. To transpose $2x$ to the LHS, subtract $2x$ from both sides:

$3x + 4 - 2x = 2x - 1 - 2x$

Simplifying this gives $x + 4 = -1$.

3. Now, subtract $4$ from both sides to isolate $x$:

$x + 4 - 4 = -1 - 4$

This simplifies to $x = -5$.

4. Substituting $x = -5$ back into the original equation:

$3(-5) + 4 = 2(-5) - 1$

This results in $-15 + 4 = -10 - 1$, which simplifies to $-11 = -11$, verifying that $x = -5$ is indeed the correct solution.

Therefore, the assertion and reason are both true, and the reason correctly explains why the assertion holds.

Key Concept: Solving Equations, Combining like terms

a) $\frac{5}{3}$

[Solution Description]

To solve the equation $5(x - 2) + 3 = 2(x + 1) - 4$, follow these steps:

1. Expand both sides of the equation:

$$5x - 10 + 3 = 2x + 2 - 4$$

$$5x - 7 = 2x - 2$$

2. Subtract $2x$ from both sides to get all variable terms on one side:

$$5x - 2x - 7 = 2x - 2x - 2$$

$$3x - 7 = -2$$

3. Add $7$ to both sides to isolate the variable term:

$$3x - 7 + 7 = -2 + 7$$

$$3x = 5$$

4. Divide both sides by $3$ to solve for $x$:

$$x = \frac{5}{3}$$

The solution to the equation is $x = \frac{5}{3}$.

Key Concept: Solving Equations, Transposing terms

c) $-12$

[Solution Description]

To solve the equation $3x + 7 = 2x - 5$, follow these steps:

1. Subtract $2x$ from both sides to get all variable terms on one side:

$$3x - 2x + 7 = 2x - 2x - 5$$

$$x + 7 = -5$$

2. Now, subtract $7$ from both sides to isolate the variable term:

$$x + 7 - 7 = -5 - 7$$

$$x = -12$$

The solution to the equation is $x = -12$.

Key Concept: Solving Equations Having Variable on Both Sides

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To solve the equation $3x + 4 = x + 10$, follow these steps:

Step 1: Subtract $x$ from both sides to consolidate the variable terms.

$3x + 4 - x = x + 10 - x$

Step 2: Simplify both sides.

$2x + 4 = 10$

Step 3: Subtract $4$ from both sides to isolate the term with $x$.

$2x = 10 - 4$

Step 4: Simplify the right-hand side.

$2x = 6$

Step 5: Divide both sides by $2$ to solve for $x$.

$x = \frac{6}{2} = 3$

Thus, the solution is $x = 3$. The Assertion is true because subtracting $x$ from both sides is the correct first step. The Reason is also true and correctly explains why this step is necessary.

Key Concept: Solving Equations, Multiplying both sides

a) $\frac{8}{5}$

[Solution Description]

To solve the equation $\frac{4x - 3}{2} = \frac{3x + 2}{4}$, follow these steps:

1. Multiply both sides of the equation by $4$ to eliminate the denominators:

$$4 \times \left(\frac{4x - 3}{2}\right) = 4 \times \left(\frac{3x + 2}{4}\right)$$

$$2(4x - 3) = 3x + 2$$

2. Expand the left side of the equation:

$$8x - 6 = 3x + 2$$

3. Subtract $3x$ from both sides to get all variable terms on one side:

$$8x - 3x - 6 = 3x - 3x + 2$$

$$5x - 6 = 2$$

4. Add $6$ to both sides to isolate the variable term:

$$5x - 6 + 6 = 2 + 6$$

$$5x = 8$$

5. Divide both sides by $5$ to solve for $x$:

$$x = \frac{8}{5}$$

The solution to the equation is $x = \frac{8}{5}$.

Key Concept: Solving Equations Having Variable on Both Sides

c) 6

[Solution Description]

Start with the equation $3x - 5 = x + 7$. Subtract $x$ from both sides to get $3x - x - 5 = 7$, which simplifies to $2x - 5 = 7$. Add 5 to both sides to get $2x = 12$. Finally, divide both sides by 2 to find $x = 6$.

Key Concept: Solving linear equations, transposing terms, fractions

b) 2

[Solution Description]

To solve the equation $\frac{2x}{3} + \frac{1}{2} = \frac{x}{2} + \frac{5}{6}$, we first eliminate the fractions by finding a common denominator.

Step 1: Multiply every term in the equation by $6$ (the least common multiple of $2, 3,$ and $6$) to eliminate the denominators:

$6 \times \left(\frac{2x}{3}\right) + 6 \times \left(\frac{1}{2}\right) = 6 \times \left(\frac{x}{2}\right) + 6 \times \left(\frac{5}{6}\right)$

Calculating each term, we get:

$4x + 3 = 3x + 5$

Step 2: Subtract $3x$ from both sides to get all the $x$ terms on one side:

$4x - 3x + 3 = 3x - 3x + 5$

Simplifying this, we get:

$x + 3 = 5$

Step 3: Subtract $3$ from both sides to solve for $x$:

$x + 3 - 3 = 5 - 3$

Simplifying this, we get:

$x = 2$

Therefore, the solution is $x = 2$.

Key Concept: Solving Equations with Variables on Both Sides

c) -12

[Solution Description]

First, eliminate the denominators by multiplying both sides by the least common multiple of 3 and 2, which is 6:

$6 \times \left(\frac{4x + 6}{3}\right) = 6 \times \left(\frac{2x - 4}{2}\right)$

Simplify both sides:

$2(4x + 6) = 3(2x - 4)$

Expand the equations:

$8x + 12 = 6x - 12$

Subtract $6x$ from both sides:

$8x - 6x + 12 = -12$

Simplify to get:

$2x + 12 = -12$

Subtract $12$ from both sides:

$2x = -24$

Finally, divide both sides by 2:

$x = -12$

Key Concept: Solving Equations Having Variable on Both Sides

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

To solve the equation $3x + 5 = 2x - 7$, we first transpose terms involving $x$ to the left-hand side (LHS) and constants to the right-hand side (RHS). Subtracting $2x$ from both sides gives $3x - 2x + 5 = -7$, which simplifies to $x + 5 = -7$. Next, subtracting $5$ from both sides yields $x = -12$. This shows that the equation has a unique solution, confirming the Assertion. The Reason explains the method used to solve such equations by transposing variables and constants, which is correct but does not directly explain why the equation has a unique solution. Therefore, both are true, but the Reason is not the correct explanation of the Assertion.

Key Concept: Solving Linear Equations with Variables on Both Sides

b) $x = 4$

[Solution Description]

To solve the equation $4x + 6 = 2x + 14$, follow these steps:

Step 1: Subtract $2x$ from both sides to get all $x$ terms on one side.

$4x - 2x + 6 = 2x - 2x + 14$

$2x + 6 = 14$

Step 2: Subtract 6 from both sides to isolate the $x$ term.

$2x + 6 - 6 = 14 - 6$

$2x = 8$

Step 3: Divide both sides by 2 to solve for $x$.

$\frac{2x}{2} = \frac{8}{2}$

$x = 4$

Therefore, the solution is $x = 4$.

Key Concept: Solving Equations with Variables on Both Sides

c) $\frac{23}{3}$

[Solution Description]

First, expand both sides of the equation:

$5x - 15 = 2x + 8$

Next, subtract $2x$ from both sides to gather like terms:

$5x - 2x - 15 = 8$

Simplify to get:

$3x - 15 = 8$

Add $15$ to both sides:

$3x = 23$

Finally, divide both sides by 3:

$x = \frac{23}{3}$

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To simplify the given equation $\frac{3x + 4}{2} - \frac{x - 1}{4} = 5$, we first find the LCM of the denominators, which is 4. Multiplying both sides of the equation by 4:

$4 \left( \frac{3x + 4}{2} \right) - 4 \left( \frac{x - 1}{4} \right) = 4 \times 5$

Simplifying each term:

$2(3x + 4) - (x - 1) = 20$

Expanding the brackets:

$6x + 8 - x + 1 = 20$

Combining like terms:

$5x + 9 = 20$

Solving for $x$:

$5x = 11 \quad \text{and} \quad x = \frac{11}{5}$

Both Assertion and Reason are true, and Reason correctly explains how multiplying by the LCM simplifies the equation.

Key Concept: Reducing Equations to Simpler Form

c) $x = \frac{12}{5}$

[Solution Description]

To solve the equation $\frac{3x + 2}{4} + \frac{x - 1}{2} = 3$, first find the LCM of the denominators, which is 4. Multiply both sides by 4:

$4 \left( \frac{3x + 2}{4} \right) + 4 \left( \frac{x - 1}{2} \right) = 4 \times 3$

Simplify the equation:

$(3x + 2) + 2(x - 1) = 12$

Expand and combine like terms:

$3x + 2 + 2x - 2 = 12$

$5x = 12$

Finally, solve for $x$:

$x = \frac{12}{5}$

So, the solution is $x = \frac{12}{5}$.

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Multiplying both sides of an equation by the LCM of the denominators is a valid method to eliminate fractions and simplify the equation. This step converts the equation into a form with integer coefficients, which is generally easier to manipulate and solve. The reason provided correctly explains why this method is effective, as the LCM ensures all denominators are canceled out, leading to a simpler equation.

Key Concept: Reducing Equations to Simpler Form

b) $x = \frac{20}{7}$

[Solution Description]

To solve the equation, we first find the LCM of the denominators, which is 4. Multiply both sides by 4:

$4 \left( \frac{3x + 2}{4} \right) + 4 \left( \frac{2x - 1}{2} \right) = 4 \times 5$

Simplifying, we get: $(3x + 2) + 2(2x - 1) = 20$

Open the brackets: $3x + 2 + 4x - 2 = 20$

Combine like terms: $7x = 20$

Divide both sides by 7: $x = \frac{20}{7}$.

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To simplify the equation $\frac{3x + 2}{4} - \frac{x - 1}{2} = 1$, we follow these steps:

1. Identify the denominators: 4 and 2. The LCM of 4 and 2 is 4.

2. Multiply every term in the equation by 4 to eliminate the denominators:

$4 \left( \frac{3x + 2}{4} \right) - 4 \left( \frac{x - 1}{2} \right) = 4 \times 1$

3. Simplify each term:

$(3x + 2) - 2(x - 1) = 4$

4. Open the brackets:

$3x + 2 - 2x + 2 = 4$

5. Combine like terms:

$x + 4 = 4$

6. Solve for $x$:

$x = 4 - 4 = 0$

The Assertion is true as the equation is correctly simplified by multiplying by 4. The Reason is also true because multiplying by the LCM of the denominators eliminates the fractions and simplifies the equation. Moreover, the Reason correctly explains the Assertion.

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The given equation is $\frac{6x + 1}{3} + 1 = \frac{x - 3}{6}$. The denominators in the equation are 3 and 6, and their LCM is 6. Multiplying both sides of the equation by 6 eliminates the fractions, leading to a simpler form of the equation. This step is correct because it simplifies the equation without changing its solution. Therefore, the Assertion is true. The Reason is also true as multiplying both sides by the LCM of the denominators is a standard method to simplify equations containing fractions. Moreover, the Reason correctly explains why this step is valid.

Key Concept: Linear Equations

a) $\frac{13}{7}$

[Solution Description]

First, find the LCM of the denominators 2 and 4, which is 4. Multiply both sides of the equation by 4:

$4 \times \frac{3x + 4}{2} + 4 \times \frac{x - 1}{4} = 4 \times 5$

Simplify each term:

$2(3x + 4) + (x - 1) = 20$

Open the brackets:

$6x + 8 + x - 1 = 20$

Combine like terms:

$7x + 7 = 20$

Subtract 7 from both sides:

$7x = 13$

Divide both sides by 7:

$x = \frac{13}{7}$

Key Concept: Linear Equations, Simplification

a) $\frac{7}{17}$

[Solution Description]

To solve the equation $\frac{3x + 2}{5} - \frac{2x - 1}{3} = \frac{x + 1}{2}$, first find the LCM of the denominators 5, 3, and 2, which is 30. Multiply both sides by 30 to eliminate the denominators:

$30 \times \left(\frac{3x + 2}{5}\right) - 30 \times \left(\frac{2x - 1}{3}\right) = 30 \times \left(\frac{x + 1}{2}\right)$

Simplify each term:

$6(3x + 2) - 10(2x - 1) = 15(x + 1)$

Expand the terms:

$18x + 12 - 20x + 10 = 15x + 15$

Combine like terms:

$-2x + 22 = 15x + 15$

Transpose $15x$ to the left side and $22$ to the right side:

$-2x - 15x = 15 - 22$

Simplify:

$-17x = -7$

Divide both sides by $-17$:

$x = \frac{7}{17}$

The solution is $x = \frac{7}{17}$.

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To solve the equation $\frac{7x - 3}{2} + \frac{1}{4} = \frac{2x + 5}{4}$, we multiply both sides by 4 to eliminate the denominators:

$4 \left( \frac{7x - 3}{2} \right) + 4 \times \frac{1}{4} = 4 \left( \frac{2x + 5}{4} \right)$

Simplifying each term:

$2(7x - 3) + 1 = 2x + 5$

Opening the brackets:

$14x - 6 + 1 = 2x + 5$

Combining like terms:

$14x - 5 = 2x + 5$

Transposing $2x$ and $-5$:

$14x - 2x = 5 + 5$

Simplifying further:

$12x = 10$

Solving for $x$:

$x = \frac{10}{12} = \frac{5}{6}$

The assertion states that the equation simplifies to $14x - 6 + 1 = 2x + 5$, which is correct after multiplying by 4 and simplifying. The reason explains that multiplying by 4 eliminates the denominators, which is also correct and directly leads to the simplification asserted.

Key Concept: Linear Equations, Simplification

a) $-\frac{5}{7}$

[Solution Description]

To solve the equation $\frac{2x + 1}{3} + \frac{3x - 1}{4} = \frac{5x - 2}{6}$, first find the LCM of the denominators 3, 4, and 6, which is 12. Multiply both sides by 12 to eliminate the denominators:

$12 \times \left(\frac{2x + 1}{3}\right) + 12 \times \left(\frac{3x - 1}{4}\right) = 12 \times \left(\frac{5x - 2}{6}\right)$

Simplify each term:

$4(2x + 1) + 3(3x - 1) = 2(5x - 2)$

Expand the terms:

$8x + 4 + 9x - 3 = 10x - 4$

Combine like terms:

$17x + 1 = 10x - 4$

Transpose $10x$ to the left side and $1$ to the right side:

$17x - 10x = -4 - 1$

Simplify:

$7x = -5$

Divide both sides by 7:

$x = -\frac{5}{7}$

The solution is $x = -\frac{5}{7}$.

Key Concept: Reducing Equations Using LCM

b) $x = -\frac{7}{3}$

[Solution Description]

The LCM of the denominators 5 and 10 is 10. Multiply every term in the equation by 10:

$10 \left( \frac{2x + 3}{5} \right) = 10 \left( \frac{x - 1}{10} \right)$

Simplifying each term:

$2(2x + 3) = x - 1$

Open the brackets:

$4x + 6 = x - 1$

Subtract $x$ from both sides:

$3x + 6 = -1$

Subtract 6 from both sides:

$3x = -7$

Divide both sides by 3:

$x = -\frac{7}{3}$

Therefore, the solution is $x = -\frac{7}{3}$.

Key Concept: Reducing Equations to Simpler Form, Use of LCM to eliminate fractions

b) 10

[Solution Description]

To solve the equation $\frac{3x + 2}{4} - \frac{2x - 1}{3} = \frac{x}{6}$, first find the LCM of the denominators 4, 3, and 6. The LCM is 12. Multiply both sides of the equation by 12 to eliminate the fractions:

$12 \left( \frac{3x + 2}{4} \right) - 12 \left( \frac{2x - 1}{3} \right) = 12 \left( \frac{x}{6} \right)$

Simplify each term:

$3(3x + 2) - 4(2x - 1) = 2x$

Expand the terms:

$9x + 6 - 8x + 4 = 2x$

Combine like terms:

$x + 10 = 2x$

Subtract x from both sides:

$10 = x$

Therefore, the solution is $x = 10$.

Key Concept: Reducing Equations to Simpler Form, Use of LCM to eliminate fractions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] The given equation is $\frac{3x + 2}{4} + \frac{x - 1}{2} = \frac{5x - 3}{6}$. The denominators are 4, 2, and 6. The LCM of 4, 2, and 6 is 12. Multiplying both sides of the equation by 12, we get:

$12 \left( \frac{3x + 2}{4} \right) + 12 \left( \frac{x - 1}{2} \right) = 12 \left( \frac{5x - 3}{6} \right)$

Simplifying each term:

$3(3x + 2) + 6(x - 1) = 2(5x - 3)$

Expanding the terms:

$9x + 6 + 6x - 6 = 10x - 6$

Combining like terms:

$15x = 10x - 6$

Solving for $x$:

$15x - 10x = -6$

$5x = -6$

$x = -\frac{6}{5}$

The assertion is true as multiplying by the LCM simplifies the equation. The reason is also true since multiplying by the LCM eliminates the fractions, simplifying the equation. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Reducing Equations Using LCM

c) $x = 3$

[Solution Description]

To solve the equation, first find the LCM of the denominators 4 and 2, which is 4. Multiply every term in the equation by 4 to eliminate the fractions:

$4 \left( \frac{3x - 1}{4} + 2 \right) = 4 \left( \frac{x + 5}{2} \right)$

Simplifying each term:

$(3x - 1) + 8 = 2(x + 5)$

Open the brackets:

$3x - 1 + 8 = 2x + 10$

Combine like terms:

$3x + 7 = 2x + 10$

Subtract $2x$ from both sides:

$x + 7 = 10$

Subtract 7 from both sides:

$x = 3$

Therefore, the solution is $x = 3$.

Key Concept: Use of LCM to eliminate fractions

c) $1$

[Solution Description]

To solve the equation, we first find the LCM of the denominators 5, 10, and 2, which is 10. We then multiply both sides of the equation by 10 to eliminate the fractions:

$10 \left( \frac{4x - 3}{5} \right) + 10 \left( \frac{2x + 1}{10} \right) = 10 \left( \frac{3x - 2}{2} \right)$

Simplifying each term:

$2(4x - 3) + (2x + 1) = 5(3x - 2)$

Expanding the brackets:

$8x - 6 + 2x + 1 = 15x - 10$

Combining like terms:

$10x - 5 = 15x - 10$

Subtracting $15x$ from both sides:

$-5x - 5 = -10$

Adding 5 to both sides:

$-5x = -5$

Dividing both sides by $-5$:

$x = 1$

Key Concept: Handling brackets, distributed terms

a) 0

[Solution Description]

To solve the equation, first eliminate the denominators by multiplying both sides by 6:

$6 \left( \frac{3x - 4}{2} \right) + 6 \left( \frac{5(2x + 1)}{3} \right) = 6 \left( \frac{7x - 2}{6} \right)$

Simplify each term:

$3(3x - 4) + 2 \cdot 5(2x + 1) = 7x - 2$

Open the brackets:

$9x - 12 + 20x + 10 = 7x - 2$

Combine like terms:

$29x - 2 = 7x - 2$

Subtract $7x$ from both sides:

$22x - 2 = -2$

Add 2 to both sides:

$22x = 0$

Divide by 22:

$x = 0$

The solution is $x = 0$.

Key Concept: Handling brackets and distributed terms

b) $a = 15$

[Solution Description]

Step 1: Expand the left-hand side by opening the bracket:

$4(a - 3) + 2 = 4a - 12 + 2 = 4a - 10$

Step 2: The equation now becomes:

$4a - 10 = 3a + 5$

Step 3: Transpose $3a$ to LHS and $-10$ to RHS:

$4a - 3a = 5 + 10$

Step 4: Simplify:

$a = 15$

Key Concept: Reducing Equations to Simpler Form, Handling brackets and distributed terms

d) Assertion is false, but Reason is true.

[Solution Description]

Step 1: Multiply both sides by 12 to eliminate the denominators:

$12 \left( \frac{3x + 2}{4} - \frac{2x - 1}{3} \right) = 12 \left( \frac{x + 5}{6} \right)$

Step 2: Simplify each term:

$3(3x + 2) - 4(2x - 1) = 2(x + 5)$

Step 3: Expand the brackets:

$9x + 6 - 8x + 4 = 2x + 10$

Step 4: Combine like terms:

$x + 10 = 2x + 10$

Step 5: Transpose terms to isolate $x$:

$x - 2x = 10 - 10$

Step 6: Solve for $x$:

$-x = 0 \implies x = 0$

Conclusion: The Assertion is false because the correct solution is $x = 0$, not $x = 7$. However, the Reason is true as multiplying by the LCM of the denominators is a valid method to simplify the equation.

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] To solve the equation $3(x - 2) = 2x + 5$, follow these steps:

1. Expand the brackets on the left-hand side: $3x - 6 = 2x + 5$.

2. Subtract $2x$ from both sides to isolate the variable: $3x - 2x - 6 = 5$, which simplifies to $x - 6 = 5$.

3. Add $6$ to both sides to solve for $x$: $x = 11$.

The Assertion is true as $x = 11$ is the correct solution. The Reason is also true because simplifying the equation indeed involves expanding the brackets and combining like terms. Moreover, the Reason correctly explains the process that leads to the Assertion.

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Step 1: Start with the given equation $3(x + 2) = 15$.

Step 2: Divide both sides of the equation by 3 to simplify it. This gives $x + 2 = 5$.

Step 3: Subtract 2 from both sides to isolate $x$. This results in $x = 3$.

Step 4: The assertion that solving the equation gives $x = 3$ is true. The reason provided correctly explains the steps taken to solve the equation.

Key Concept: Reducing Equations to Simpler Form

b) $-7$

[Solution Description]

Eliminate the denominators by multiplying both sides by 6 (the LCM of 3 and 2):

$6 \times \left( \frac{2x + 5}{3} \right) = 6 \times \left( \frac{x + 1}{2} \right)$

Simplifying both sides, we get:

$2(2x + 5) = 3(x + 1)$

Expanding both sides:

$4x + 10 = 3x + 3$

Transposing $3x$ to the left side and $10$ to the right side:

$4x - 3x = 3 - 10$

Simplifying further:

$x = -7$

Therefore, the solution is $x = -7$.

Key Concept: Reducing Equations to Simpler Form

c) $-4$

[Solution Description]

To solve the equation, first eliminate the denominators by multiplying both sides by 4 (the LCM of 4 and 2). This gives:

$4 \times \left( \frac{3x + 2}{4} \right) = 4 \times \left( \frac{x - 1}{2} \right)$

Simplifying both sides, we get:

$3x + 2 = 2(x - 1)$

Expanding the right side:

$3x + 2 = 2x - 2$

Transposing $2x$ to the left side and $2$ to the right side:

$3x - 2x = -2 - 2$

Simplifying further:

$x = -4$

Therefore, the solution is $x = -4$.

Key Concept: Reducing Equations to Simpler Form, Stepwise approach

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To simplify the given equation, multiply both sides by the LCM of the denominators, which is 4. This eliminates the fractions and simplifies the equation:

$4 \left( \frac{3x - 1}{2} \right) + 4 \left( \frac{x + 2}{4} \right) = 4 \left( \frac{5x - 3}{4} \right)$

Simplifying each term:

$2(3x - 1) + (x + 2) = 5x - 3$

Expanding the terms:

$6x - 2 + x + 2 = 5x - 3$

Combining like terms:

$7x = 5x - 3$

Transposing $5x$ to the left side:

$2x = -3$

Dividing by 2:

$x = -\frac{3}{2}$

Both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To simplify the equation $\frac{3x + 2}{4} + \frac{1}{4} = \frac{x - 1}{4}$, we can multiply both sides by 4 to eliminate the denominator. This is a standard technique used to reduce equations to simpler forms. After multiplication, the equation becomes $3x + 2 + 1 = x - 1$, which simplifies further to $3x + 3 = x - 1$. Solving this, we get $2x = -4$ and $x = -2$. Therefore, the assertion is true, and the reason correctly explains why multiplying by 4 simplifies the equation.

Key Concept: Reducing Equations to Simpler Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let us verify the assertion and reason step by step. Start with the equation $\frac{2x + 3}{4} = \frac{x - 1}{2}$. The LCM of the denominators 4 and 2 is 4. Multiply both sides of the equation by 4: $4 \times \frac{2x + 3}{4} = 4 \times \frac{x - 1}{2}$. Simplifying, we get $2x + 3 = 2(x - 1)$. This proves that the assertion is true. The Reason states that multiplying both sides of an equation by the LCM of the denominators eliminates fractions and simplifies the equation, which is also true. Further, the Reason correctly explains why the Assertion is valid. Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Ages, Numbers

c) 24

[Solution Description]

Let the current age of the son be $x$. Then, the current age of the father is $65 - x$. Five years ago, the son's age was $x - 5$ and the father's age was $(65 - x) - 5 = 60 - x$. According to the problem:

$60 - x = 2(x - 5)$

Expand the right side:

$60 - x = 2x - 10$

Add $x$ to both sides:

$60 = 3x - 10$

Add 10 to both sides:

$70 = 3x$

Divide by 3:

$x = \frac{70}{3} \approx 23.33$

Since age must be an integer, there is no valid solution with this condition.

Key Concept: Linear Equations, Ages

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] Let the son's age be $x$ years. Then the father's age is $x + 30$ years. According to the problem, the sum of their ages is 50 years. So, we can write the equation as: $x + (x + 30) = 50$ Simplifying this, we get: $2x + 30 = 50$ Subtracting 30 from both sides: $2x = 20$ Dividing both sides by 2: $x = 10$ Therefore, the son's age is 10 years. The assertion is true. The reason correctly explains how to solve the problem using a linear equation, making it a valid explanation.

Key Concept: Applications of Linear Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] Let the current age of the son be $x$. According to the assertion, the father's current age is $3x$. After 15 years, the son's age will be $x + 15$, and the father's age will be $3x + 15$. The assertion states that after 15 years, the father will be twice as old as his son, which translates to the equation $3x + 15 = 2(x + 15)$. Solving this equation: First, expand the right-hand side: $3x + 15 = 2x + 30$. Next, subtract $2x$ from both sides: $x + 15 = 30$. Then, subtract $15$ from both sides: $x = 15$. Thus, the son's current age is $15$ years, and the father's current age is $45$ years. The reason correctly forms the linear equation based on the assertion, and the reason is the correct explanation for the assertion.

Key Concept: Perimeter

c) $\frac{20}{3}$ cm and $\frac{55}{3}$ cm

[Solution Description]

Let the width be $w$. Then the length is $2w + 5$. The perimeter of a rectangle is given by:

$2(\text{length} + \text{width}) = 50$

Substituting the length and width:

$2(2w + 5 + w) = 50$

$2(3w + 5) = 50$

$6w + 10 = 50$

$6w = 40$

$w = \frac{40}{6} = \frac{20}{3}$

The length is:

$2\left(\frac{20}{3}\right) + 5 = \frac{40}{3} + 5 = \frac{40}{3} + \frac{15}{3} = \frac{55}{3}$

Therefore, the dimensions are $\frac{20}{3}$ cm and $\frac{55}{3}$ cm.

Key Concept: Applications of Linear Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] To find the width $w$, we use the perimeter formula for a rectangle: $P = 2(l + w)$. Given $P = 20$ units and $l = 6$ units, substituting these values into the formula gives $2(6 + w) = 20$. Simplifying this equation step by step:

$2(6 + w) = 20 \\ \\ 6 + w = \frac{20}{2} \\ \\ 6 + w = 10 \\ \\ w = 10 - 6 \\ \\ w = 4$

Therefore, the width $w$ is 4 units. Both the assertion and reason are true, and the reason correctly explains the assertion through the application of the perimeter formula.

Key Concept: Linear Equations, Word Problems on Numbers

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] Let the first number be $x$ and the second number be $y$. According to the problem, $x = 2y + 6$. Also, $x + y = 24$. We can substitute the expression for $x$ from the first equation into the second equation: $2y + 6 + y = 24$. Simplifying this, we get $3y + 6 = 24$. Subtracting 6 from both sides, $3y = 18$. Dividing both sides by 3, $y = 6$. Substituting $y = 6$ back into the first equation, $x = 2(6) + 6 = 12 + 6 = 18$. Therefore, the larger number is indeed 18. Both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Word problems on numbers

b) 7

[Solution Description]

Let the number be $x$. According to the problem, $3x - 5 = 16$. Adding 5 to both sides of the equation, we get $3x = 21$. Dividing both sides by 3, we find $x = 7$. Therefore, the number is 7.

Key Concept: Word problems on numbers

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description] Let the two consecutive integers be $x$ and $x+1$. According to the problem, their sum is 25. Therefore, we can write the equation as:

$x + (x + 1) = 25$

Simplifying, we get:

$2x + 1 = 25$

Subtracting 1 from both sides:

$2x = 24$

Dividing both sides by 2:

$x = 12$

Thus, the larger integer is $x + 1 = 13$. The assertion is correct. The reason states that the difference between two consecutive integers is always 1, which is true but does not directly explain why the larger integer is 13. Hence, both the assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Word problem on numbers

c) 30 and 15

[Solution Description]

Let the two numbers be $x$ and $y$. According to the problem, we have two equations:

$x + y = 45$

$x - y = 15$

Adding both equations gives us:

$2x = 60$

Solving for $x$:

$x = 30$

Substituting $x = 30$ in the first equation:

$30 + y = 45$

Solving for $y$:

$y = 15$

Therefore, the two numbers are 30 and 15.

Key Concept: Age-related problems

a) 12

[Solution Description]

Let the current age of the daughter be $x$ and the man be $y$. According to the problem:

$y = 4x$ (Equation 1)

In six years, their ages will be $(x + 6)$ and $(y + 6)$, respectively. According to the second condition:

$y + 6 = 3(x + 6)$ (Equation 2)

Substituting Equation 1 into Equation 2:

$4x + 6 = 3x + 18$

Solving for $x$:

$x = 12$

Thus, the daughter's current age is 12 years.

Key Concept: Age-related problems, Linear Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let the son's age be $S$. According to the problem, the father's age is $4S$. The sum of their ages is given as 50 years. So, we can write the equation:

$S + 4S = 50$

Simplifying the equation:

$5S = 50$

Solving for $S$:

$S = 10$

Therefore, the son's age is 10 years. The assertion is true. Now, the reason states that the father's age can be calculated using the equation $F = 4S$, which is also true because it correctly represents the relationship between the father's and son's ages. Moreover, the reason correctly explains why the assertion is true since it provides the equation used to derive the son's age.

Key Concept: Age-related problems

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that doubling a person's age will always result in a greater value than their current age. This is true because when you double any positive number, the result is always greater than the original number. The reason given is also correct: doubling a positive number always results in a larger number. Furthermore, the reason correctly explains why the assertion is true. Therefore, both the assertion and the reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Future age

c) 20 years

[Solution Description]

Let the number of years required be $x$. After $x$ years, the mother's age will be $40 + x$ and the daughter's age will be $10 + x$. According to the problem:

$40 + x = 2(10 + x)$

Expanding the right side:

$40 + x = 20 + 2x$

Subtracting $x$ from both sides:

$40 = 20 + x$

Subtracting 20 from both sides:

$x = 20$

So, the mother will be twice as old as her daughter in 20 years.

Key Concept: Perimeter of a rectangle

b) 5 cm

[Solution Description]

Let the width of the rectangle be $x$ cm. Since the length is twice the width, the length is $2x$ cm. The perimeter of a rectangle is given by the formula: $P = 2(\text{length} + \text{width})$. Substituting the given values: $30 = 2(2x + x)$. Simplifying: $30 = 2(3x)$, so $30 = 6x$. Dividing both sides by 6 gives $x = 5$. Therefore, the width of the rectangle is 5 cm.

Key Concept: Perimeter of a square

b) 6 cm

[Solution Description]

The perimeter of a square is given by the formula: $P = 4 \times \text{side length}$. Let the side length be $x$ cm. Substituting the given perimeter: $24 = 4x$. Dividing both sides by 4 gives $x = 6$. Therefore, the length of one side of the square is 6 cm.

Key Concept: Perimeter-based linear equations

b) 8 cm

[Solution Description]

First, calculate the perimeter of the rectangle using the formula $P = 2(\text{length} + \text{width})$.

$P = 2(10 + 6) = 2(16) = 32 \text{ cm}$

Since the perimeter of the square is also 32 cm, and the perimeter of a square is given by $P = 4 \times \text{side length}$, we can set up the equation:

$4s = 32$

Solve for $s$:

$s = \frac{32}{4} = 8 \text{ cm}$

The side length of the square is 8 cm.

Key Concept: Perimeter-based problems, Linear Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let's analyze both the Assertion and Reason:

1. **Assertion Analysis**: The perimeter of a rectangle is calculated using the formula $P = 2(l + w)$. If the perimeter is increased by 10 units, the new perimeter becomes $P_{\text{new}} = P + 10$. Since 10 is a positive number, $P_{\text{new}}$ will always be greater than $P$, regardless of the values of $l$ and $w$.

2. **Reason Analysis**: The Reason states that increasing any positive quantity by 10 units will result in a larger value. This is mathematically correct because adding a positive number to any positive quantity will always increase its value.

3. **Relationship Between Assertion and Reason**: The Reason correctly explains why the Assertion is true. The increase in perimeter is directly due to the addition of a positive quantity (10 units), which aligns with the Reason provided.

Key Concept: Currency Conversion

c) 127.5 EUR

[Solution Description]

To convert USD to EUR, multiply the amount in USD by the exchange rate.

Given the exchange rate is 1 USD = 0.85 EUR, the conversion for 150 USD is:

$150 \times 0.85 = 127.5$

Therefore, you will get 127.5 Euros for 150 USD.

Key Concept: Money-related problems, Linear equations

c) 6

[Solution Description]

Let the number of type A pens be $x$ and the number of type B pens be $y$. We have the following system of equations:

$x + y = 10$

$2x + 3y = 24$

From the first equation, express $y$ in terms of $x$:

$y = 10 - x$

Substitute $y$ into the second equation:

$2x + 3(10 - x) = 24$

Simplify the equation:

$2x + 30 - 3x = 24$

Combine like terms:

$-x + 30 = 24$

Subtract 30 from both sides:

$-x = -6$

Multiply both sides by -1:

$x = 6$

Key Concept: Applications of Linear Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] Let the smaller part be $x$. According to the problem, the larger part is twice the smaller part, so it is $2x$. The total amount is \$20, so we can write the equation $x + 2x = 20$. Simplifying this, we get $3x = 20$. Dividing both sides by 3, we find $x = \frac{20}{3} \approx 6.67$. Therefore, the smaller part is indeed \$6.67. Both the assertion and the reason are true, and the reason correctly explains how the assertion is derived.

Key Concept: Combination of currency notes

b) 4

[Solution Description]

Let the number of 10-dollar bills be $x$. Then, the number of 5-dollar bills is $2x$. The total amount of money can be expressed as:

$10x + 5(2x) = 50$

Simplifying the equation:

$10x + 10x = 50$

$20x = 50$

Solving for $x$:

$x = \frac{50}{20} = 2.5$

Since the number of bills must be an integer, this problem has no valid solution within the given constraints.

Key Concept: Real-life applications of linear equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] The assertion states that linear equations can be used to determine the age of a person based on given conditions. This is true because age-related problems often involve setting up a linear equation where the unknown variable represents the age. The reason explains that age-related problems involve setting up an equation where the unknown represents the age and solving for it, which is also true. Moreover, the reason correctly explains why the assertion is true because it provides the method used in such problems. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Real-life application of linear equations

c) \$35

[Solution Description]

The cost of one pen is \$5 and one notebook is \$10. The total cost for 3 pens and 2 notebooks can be calculated as follows:

$\text{Total Cost} = (3 \times 5) + (2 \times 10) = 15 + 20 = 35$

Therefore, the total cost is \$35.

Key Concept: Real-life applications of linear equations, Combination of currency notes

d) Assertion is false, but Reason is true.

[Solution Description]

Let the number of \$10 notes be $x$. Then, the number of \$5 notes is $2x$. The total value of \$10 notes is $10x$, and the total value of \$5 notes is $5(2x)$. The total value of all notes is given by $10x + 10x = 20x$. According to the problem, $20x = 50$. Solving for $x$, we get $x = \frac{50}{20} = 2.5$. However, the number of notes must be an integer, which contradicts the assertion that the total number of notes is 8. Therefore, the assertion is false. The reason correctly sets up the equation based on the given information, but the solution does not support the assertion.

Key Concept: Application of Linear Equations - Perimeter Problem

c) $\frac{44}{3}$ cm

[Solution Description]

Let the width of the rectangle be $w$ cm. Then, the length of the rectangle is $(2w + 4)$ cm. The perimeter of a rectangle is given by the formula: $\text{Perimeter} = 2(\text{length} + \text{width})$. Substituting the given values, we have: $2((2w + 4) + w) = 40$. Simplifying this, we get: $2(3w + 4) = 40$. Expanding, we get: $6w + 8 = 40$. Subtracting 8 from both sides, we get: $6w = 32$. Dividing both sides by 6, we get: $w = \frac{32}{6} = \frac{16}{3}$ cm. Now, the length is: $2\left(\frac{16}{3}\right) + 4 = \frac{32}{3} + 4 = \frac{32}{3} + \frac{12}{3} = \frac{44}{3}$ cm. Therefore, the length of the rectangle is $\frac{44}{3}$ cm.