Key Concept: Linear Equations in One Variable

c) 7

[Solution Description]

To solve the equation $3x - 7 = 14$, we need to isolate $x$. Start by adding 7 to both sides of the equation:

$3x - 7 + 7 = 14 + 7$

Simplifying, we get:

$3x = 21$

Next, divide both sides by 3:

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

So, the solution is $x = 7$.

Key Concept: Linear Equations in One Variable

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

[Solution Description]

A linear equation in one variable is an equation where the highest power of the variable is 1. The given equation $3x + 5 = 11$ is linear because the highest power of $x$ is 1. This satisfies the definition of a linear equation in one variable. Therefore, both the Assertion and Reason are true, and the Reason correctly explains why the Assertion is true.

Key Concept: Linear Expressions

b) $4x + 5$

[Solution Description]

A linear expression in one variable has the highest power of the variable equal to 1. Out of the given options, $4x + 5$ is a linear expression because the highest power of $x$ is 1.

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: Linear Equations in One Variable, Algebraic Expressions

a) $\frac{58}{19}$

[Solution Description]

First, find a common denominator for the fractions, which is 6. Multiply each term by 6 to eliminate the denominators:

$6 \cdot \left( \frac{3x - 4}{2} \right) + 6 \cdot \left( \frac{5x + 7}{3} \right) = 6 \cdot 10$

Simplify each term:

$3(3x - 4) + 2(5x + 7) = 60$

Distribute the constants:

$9x - 12 + 10x + 14 = 60$

Combine like terms:

$19x + 2 = 60$

Subtract 2 from both sides:

$19x = 58$

Divide both sides by 19:

$x = \frac{58}{19}$

Therefore, the solution is $x = \frac{58}{19}$.

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: Algebraic Expressions

b) $2x + 3$

[Solution Description]

A linear expression in one variable has only one variable with the highest power of 1. Let's analyze each option:

Option 1: $x^2 + 1$ is not linear because the power of $x$ is 2.

Option 2: $2x + 3$ is linear because the power of $x$ is 1.

Option 3: $y + y^2$ is not linear because the power of $y$ is 2.

Option 4: $xyz + x + y + z$ is not linear because it has multiple variables and powers greater than 1.

Therefore, the correct answer is $2x + 3$.

Key Concept: Definition of Algebraic Expression

a) $3x + 5$

[Solution Description]

An algebraic expression is a combination of variables and constants using mathematical operations like addition, subtraction, multiplication, and division. It does not contain an equality (\$) sign. From the given options, $3x + 5$ is an algebraic expression as it consists of a variable $x$, a constant $5$, and an addition operation. The other options either have an equality sign or are not expressions.

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: 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: Linear Equations in One Variable

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. Among the given options, $2x + 3 = 7$ meets these criteria.

Key Concept: Difference between expressions and equations

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

[Solution Description] An equation must contain an equality sign ($=$). Among the given options, only option (d) contains an equality sign, making it an equation. The other options are expressions as they do not have an equality sign.

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: 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: Linear Expressions, Equations

c) Only $4y - 7$ can be used.

[Solution Description]

A linear equation in one variable is formed by setting a linear expression (where the highest power of the variable is 1) equal to another expression or number. $4y - 7$ is a linear expression because the highest power of $y$ is 1. However, $y^2 + 3$ is not linear because the highest power of $y$ is 2. Therefore, only $4y - 7$ can be used to form a linear equation in one variable.

Key Concept: Left Hand Side and Right Hand Side

a) $5x - 10$

[Solution Description]

The Left Hand Side (LHS) of an equation is the expression on the left of the equality sign. In the equation $5x - 10 = 15$, the LHS is $5x - 10$.

Key Concept: Equations vs Expressions

b) $2x - 3 = 9$

[Solution Description]

An equation must have an equality sign ($=$). Among the given options, $2x - 3 = 9$ is an equation because it contains an equality sign.

Key Concept: Linear Expressions

b) $3x + 5$

[Solution Description]

A linear expression in one variable has the highest power of the variable as 1. Among the given options, $3x + 5$ is a linear expression because the highest power of $x$ is 1.

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, Variables and Constants

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

[Solution Description]

The assertion states that the equation $3x + 5 = 2x + 10$ is a linear equation in one variable. This is true because it satisfies the definition of a linear equation in one variable, which requires the equation to have only one variable and the highest power of that variable to be 1.

The reason states that linear equations in one variable are those where the highest power of the variable is 1. This is also true as per the definition of linear equations in one variable.

Furthermore, the reason correctly explains why the equation in the assertion is a linear equation in one variable. Therefore, both the assertion and the reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Linear Equations in One Variable

a) $3x - 5 = 0$

[Solution Description]

A linear equation in one variable has the form $ax + b = 0$, where $a$ and $b$ are constants, and the highest power of the variable $x$ is 1. The equation $3x - 5 = 0$ fits this form because it contains only one variable $x$ with its highest power as 1. Therefore, it is a linear equation in one variable.

Key Concept: Linear Equations

a) $3x + 5 = 11$

[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. Let us check each option:

- $3x + 5 = 11$ is a linear equation because the highest power of $x$ is 1.

- $x^2 + 2x = 4$ is not a linear equation because the highest power of $x$ is 2.

- $y + 3 = 7$ is a linear equation but it has a different variable $y$.

- $2xy = 10$ is not a linear equation because it has two variables $x$ and $y$.

Therefore, the correct answer is $3x + 5 = 11$.

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 Expressions

b) $4z + 7$

[Solution Description]

A linear expression has the highest power of the variable as 1. The expression $4z + 7$ is a linear expression because the highest power of $z$ is 1. Other expressions like $z^2 + 3$ and $z + z^3$ have powers greater than 1, so they are not linear.

Key Concept: Linear Equations, Algebraic Expressions

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

[Solution Description]

To determine if the Assertion and Reason are correct, let's analyze them step by step.

**Step 1:** Identify the type of equation given in the Assertion.

The equation provided is $3x + 5 = 2x - 1$. This equation contains only one variable, which is $x$.

**Step 2:** Check the highest power of the variable in the equation.

In the equation $3x + 5 = 2x - 1$, the highest power of $x$ is 1. This satisfies the condition for a linear equation in one variable.

**Step 3:** Verify the Reason statement.

The Reason states that a linear equation in one variable has the highest power of the variable as 1. This is a correct definition of a linear equation in one variable.

**Conclusion:** Both the Assertion and Reason are true, and the Reason correctly explains why the equation in the Assertion is linear.

Key Concept: Solving Equations with Variables on Both Sides

a) No Solution

[Solution Description]

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

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

Simplifying, we have:

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

Expand the right side:

$2x + 3 = 2x - 2$

Subtract $2x$ from both sides:

$3 = -2$

This is a contradiction, which means there is no solution to the equation.

Key Concept: Variable on Both Sides, Brackets

b) $x = 2$

[Solution Description]

Start with the equation:

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

Expand both sides:

$3x - 12 + 2 = 5x - 6x - 2$

Simplify both sides:

$3x - 10 = -x - 2$

Add $x$ to both sides:

$3x + x - 10 = -2$

Simplify to obtain:

$4x - 10 = -2$

Add $10$ to both sides:

$4x = -2 + 10$

Simplify to obtain:

$4x = 8$

Divide both sides by 4:

$x = 2$

Key Concept: Variable on Both Sides, Transposition

c) $x = 5$

[Solution Description]

Start with the equation:

$3x + 4 = 2x + 9$

Subtract $2x$ from both sides to get:

$3x - 2x + 4 = 9$

Simplify to obtain:

$x + 4 = 9$

Now, subtract $4$ from both sides:

$x = 9 - 4$

Finally, we get:

$x = 5$

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: 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: Balancing Equations

b) $x = 1$

[Solution Description]

Multiply both sides of the equation by $2$ to eliminate the denominators: $4x + 6 = 2x + 8$. Subtract $2x$ from both sides: $4x - 2x + 6 = 8$, which simplifies to $2x + 6 = 8$. Subtract $6$ from both sides: $2x = 8 - 6$, so $2x = 2$. Divide both sides by $2$: $x = 1$.

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

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

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: Balancing Equations, Combining Like Terms

c) 12

[Solution Description]

To solve the equation $4x - 7 = 3x + 5$, we first transpose the $3x$ term to the left side and the constant term $-7$ to the right side:

$4x - 3x = 5 + 7$

Simplifying both sides:

$x = 12$

Key Concept: Transposing terms

b) $-9$

[Solution Description]

To solve the equation $3x + 4 = 2x - 5$, we need to transpose the terms involving $x$ to one side and the constant terms to the other side.

Step 1: Subtract $2x$ from both sides of the equation:

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

This simplifies to:

$x + 4 = -5$

Step 2: Subtract $4$ from both sides:

$x = -5 - 4$

$x = -9$

Thus, the solution to the equation is $x = -9$.

Key Concept: Transposing terms

b) $3$

[Solution Description]

To solve the equation $4x + 6 = 2x + 12$, we need to transpose the terms involving $x$ to one side and the constant terms to the other side.

Step 1: Subtract $2x$ from both sides of the equation:

$4x - 2x + 6 = 12$

This simplifies to:

$2x + 6 = 12$

Step 2: Subtract $6$ from both sides:

$2x = 12 - 6$

$2x = 6$

Step 3: Divide both sides by $2$:

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

$x = 3$

Thus, the solution to the equation is $x = 3$.

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 Linear Equations

a) -7

[Solution Description]

To solve the equation $\frac{4x + 6}{2} = \frac{2x - 8}{2}$, we first multiply both sides by 2 to eliminate the denominators:

$2 \times \left(\frac{4x + 6}{2}\right) = 2 \times \left(\frac{2x - 8}{2}\right)$. This simplifies to:

$4x + 6 = 2x - 8$. Next, subtract $2x$ from both sides:

$4x - 2x + 6 = 2x - 2x - 8$. This simplifies to:

$2x + 6 = -8$. Then, subtract 6 from both sides:

$2x = -8 - 6$. This gives:

$2x = -14$. Finally, divide both sides by 2 to find:

$x = \frac{-14}{2}$. Therefore, the solution is:

$x = -7$.

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 Equations with Variables on Both Sides

a) $10$

[Solution Description]

To solve the equation $5x - 9 = 3x + 11$, we need to isolate the variable $x$ on one side of the equation. Here are the steps:

1. Subtract $3x$ from both sides:

$5x - 9 - 3x = 3x + 11 - 3x$

This simplifies to:

$2x - 9 = 11$

2. Add $9$ to both sides:

$2x - 9 + 9 = 11 + 9$

This gives us:

$2x = 20$

3. Divide both sides by 2:

$\frac{2x}{2} = \frac{20}{2}$

This gives us:

$x = 10$

Therefore, the solution is $x = 10$.

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 $3x + 4 = x + 10$, 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 Having Variable on Both Sides

b) 4

[Solution Description]

Start with the equation $5x + 3 = 2x + 15$. Subtract $2x$ from both sides to get $3x + 3 = 15$. Subtract 3 from both sides to get $3x = 12$. Finally, divide both sides by 3 to find $x = 4$.

Key Concept: Solving Equations Having Variable on Both Sides

b) -7

[Solution Description]

Start with the equation $\frac{4x + 6}{2} = \frac{2x - 8}{2}$. Multiply both sides by 2 to eliminate the denominators: $4x + 6 = 2x - 8$. Subtract $2x$ from both sides to get $2x + 6 = -8$. Subtract 6 from both sides to get $2x = -14$. Finally, divide both sides by 2 to find $x = -7$.

Key Concept: Solving Equations with Variables on Both Sides

a) $-12$

[Solution Description]

To solve the equation $3x + 5 = 2x - 7$, we need to isolate the variable $x$ on one side of the equation. Here are the steps:

1. Subtract $2x$ from both sides:

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

This simplifies to:

$x + 5 = -7$

2. Subtract $5$ from both sides:

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

This gives us:

$x = -12$

Therefore, the solution is $x = -12$.

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 Linear Equations with Variables on Both Sides

c) $x = 5$

[Solution Description]

To solve the equation $5x - 7 = 2x + 8$, follow these steps:

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

$5x - 2x - 7 = 2x - 2x + 8$

$3x - 7 = 8$

Step 2: Add 7 to both sides to isolate the $x$ term.

$3x - 7 + 7 = 8 + 7$

$3x = 15$

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

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

$x = 5$

Therefore, the solution 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]

Let us solve the equation step by step. The given equation is $3x - 5 = x + 7$. First, subtract $x$ from both sides:

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

Simplifying this, we get:

$2x - 5 = 7$

Next, add $5$ to both sides:

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

This simplifies to:

$2x = 12$

Finally, divide both sides by $2$:

$x = \frac{12}{2} = 6$

Thus, the solution is $x = 6$.

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 linear equations, transposing terms

c) 9

[Solution Description]

To solve the equation $3x - 4 = 2x + 5$, we need to isolate the variable $x$.

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

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

Simplifying this, we get:

$x - 4 = 5$

Step 2: Add $4$ to both sides to solve for $x$:

$x - 4 + 4 = 5 + 4$

Simplifying this, we get:

$x = 9$

Therefore, the solution is $x = 9$.

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

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

[Solution Description]

To solve the equation, first find the LCM of the denominators (2, 4, and 8), which is 8. Multiply both sides by 8 to eliminate the fractions:

$8 \left( \frac{5x - 3}{2} \right) - 8 \left( \frac{2x + 1}{4} \right) = 8 \left( \frac{3x - 7}{8} \right)$

Simplifying each term:

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

Expand the brackets:

$20x - 12 - 4x - 2 = 3x - 7$

Combine like terms:

$16x - 14 = 3x - 7$

Move all $x$ terms to one side and constants to the other:

$16x - 3x = -7 + 14$

Simplify:

$13x = 7$

Divide both sides by 13:

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

The solution is $x = \frac{7}{13}$.

Key Concept: Reducing Equations to Simpler Form

b) $x = \frac{24}{7}$

[Solution Description]

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

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

Simplifying, we get: $2(5x - 3) - (3x + 4) = 14$

Open the brackets: $10x - 6 - 3x - 4 = 14$

Combine like terms: $7x - 10 = 14$

Add 10 to both sides: $7x = 24$

Divide both sides by 7: $x = \frac{24}{7}$.

Key Concept: Reducing Equations to Simpler Form

b) $x = 2$

[Solution Description]

To solve the equation $\frac{5x - 2}{3} + \frac{2x + 1}{6} = \frac{7}{2}$, first find the LCM of the denominators, which is 6. Multiply both sides by 6:

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

Simplify the equation:

$2(5x - 2) + (2x + 1) = 21$

Expand and combine like terms:

$10x - 4 + 2x + 1 = 21$

$12x - 3 = 21$

Move the constant term to the other side:

$12x = 21 + 3$

$12x = 24$

Finally, solve for $x$:

$x = \frac{24}{12} = 2$

So, the solution is $x = 2$.

Key Concept: Reducing Equations to Simpler Form

b) $x = 4$

[Solution Description]

To solve the equation, we first find the LCM of the denominators, which is 6. Multiply both sides by 6:

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

Simplifying, we get: $2(2x + 5) - (x - 2) = 24$

Open the brackets: $4x + 10 - x + 2 = 24$

Combine like terms: $3x + 12 = 24$

Subtract 12 from both sides: $3x = 12$

Divide both sides by 3: $x = 4$.

Key Concept: Reducing Equations to Simpler Form

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

[Solution Description]

To solve the equation, first eliminate the denominators by multiplying each term by 15:

$15 \left( \frac{4x - 3}{5} \right) - 15 \times 1 = 15 \left( \frac{2x + 1}{3} \right)$

Simplify:

$3(4x - 3) - 15 = 5(2x + 1)$

Expand the brackets:

$12x - 9 - 15 = 10x + 5$

Combine like terms:

$12x - 24 = 10x + 5$

Move all $x$ terms to one side and constants to the other:

$12x - 10x = 5 + 24$

Simplify:

$2x = 29$

Divide both sides by 2:

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

So, the solution is $x = \frac{29}{2}$.

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 $\frac{2x + 1}{3} = \frac{x - 2}{6}$, we multiply both sides by 6, which is the LCM of the denominators 3 and 6. This step eliminates the fractions:

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

Simplifying both sides:

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

Opening the brackets:

$4x + 2 = x - 2$

Transposing like terms:

$4x - x = -2 - 2$

$3x = -4$

$x = -\frac{4}{3}$

Thus, the assertion and reason are both true, and the reason correctly explains why the equation was simplified by multiplying both sides by 6.

Key Concept: Reducing Equations to Simpler Form

b) $x = \frac{3}{4}$

[Solution Description]

First, simplify both sides of the equation by expanding the brackets:

$2x - 6x + 15 = 12x - 4 + 7$

Combine like terms on both sides:

$-4x + 15 = 12x + 3$

Move all $x$ terms to one side and constants to the other:

$-4x - 12x = 3 - 15$

Simplify:

$-16x = -12$

Divide both sides by $-16$:

$x = \frac{-12}{-16} = \frac{3}{4}$

So, the solution is $x = \frac{3}{4}$.

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: Use of LCM to eliminate fractions

c) $\frac{3}{4}$

[Solution Description]

To solve the equation, we first find the LCM of the denominators 3, 6, and 2, which is 6. We then multiply both sides of the equation by 6 to eliminate the fractions:

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

Simplifying each term:

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

Expanding the brackets:

$4x + 2 + x - 2 = 9x - 3$

Combining like terms:

$5x = 9x - 3$

Subtracting $9x$ from both sides:

$-4x = -3$

Dividing both sides by $-4$:

$x = \frac{3}{4}$

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 to Simpler Form

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

[Solution Description]

The assertion states that to solve the equation $\frac{3x + 2}{4} - \frac{x - 1}{3} = 1$, we must multiply both sides by the LCM of the denominators, which is 12. This is correct because the LCM of 4 and 3 is indeed 12. Multiplying both sides by 12 will eliminate the fractions, simplifying the equation to $3(3x + 2) - 4(x - 1) = 12$. The reason explains why multiplying by the LCM of the denominators is necessary: it eliminates fractions and simplifies the equation. Since both the assertion and reason are true, and the reason correctly explains the assertion, the correct answer is option a).

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

b) $\frac{1}{3}$

[Solution Description]

To solve the equation $\frac{2x + 3}{5} + \frac{3x - 4}{2} = \frac{7x}{10} - 1$, first find the LCM of the denominators 5, 2, and 10. The LCM is 10. Multiply both sides of the equation by 10 to eliminate the fractions:

$10 \left( \frac{2x + 3}{5} \right) + 10 \left( \frac{3x - 4}{2} \right) = 10 \left( \frac{7x}{10} \right) - 10 \times 1$

Simplify each term:

$2(2x + 3) + 5(3x - 4) = 7x - 10$

Expand the terms:

$4x + 6 + 15x - 20 = 7x - 10$

Combine like terms:

$19x - 14 = 7x - 10$

Subtract 7x from both sides:

$12x - 14 = -10$

Add 14 to both sides:

$12x = 4$

Divide both sides by 12:

$x = \frac{1}{3}$

Therefore, the solution is $x = \frac{1}{3}$.

Key Concept: Handling brackets and distributed terms

a) No solution

[Solution Description]

First, expand the brackets on the left side:

$3(x - 4) = 3x - 12$

$2(2x + 5) = 4x + 10$

Combine these to get:

$3x - 12 + 4x + 10 = 7x - 1$

Simplify by combining like terms:

$7x - 2 = 7x - 1$

Subtract $7x$ from both sides:

$-2 = -1$

This is not possible, so there is no solution.

Key Concept: Handling brackets and distributed terms

c) $x = 7$

[Solution Description]

First, expand the brackets on the left side:

$4(3x - 2) = 12x - 8$

$-2(5x - 1) = -10x + 2$

Combine these to get:

$12x - 8 - 10x + 2 = 8$

Simplify by combining like terms:

$2x - 6 = 8$

Add $6$ to both sides:

$2x = 14$

Divide both sides by $2$:

$x = 7$

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, distributed terms

c) 4

[Solution Description]

Start by opening the brackets on both sides:

$4x - 6x + 15 = 2x - 8 + 7$

Combine like terms:

$-2x + 15 = 2x - 1$

Add $2x$ to both sides:

$15 = 4x - 1$

Add 1 to both sides:

$16 = 4x$

Divide by 4:

$x = 4$

The solution is $x = 4$.

Key Concept: Handling brackets and distributed terms

b) $y = \frac{9}{5}$

[Solution Description]

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

$2(3y - 1) = 6y - 2$

Step 2: The equation now becomes:

$6y - 2 = y + 7$

Step 3: Transpose $y$ to LHS and $-2$ to RHS:

$6y - y = 7 + 2$

Step 4: Simplify:

$5y = 9$

Step 5: Divide both sides by 5:

$y = \frac{9}{5}$

Key Concept: Reducing Equations to Simpler Form

a) No Solution

[Solution Description]

Start by expanding both sides of the equation:

$6x - 2 + 4 = 6x + 3 - 5$

Simplify both sides:

$6x + 2 = 6x - 2$

Subtract $6x$ from both sides:

$2 = -2$

This is a contradiction, meaning there is no solution to 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 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: LCM of Denominators, Simplifying Fractions

a) $\frac{4}{5}$

[Solution Description]

Find the LCM of the denominators (3, 4, 6), which is 12. Multiply both sides by 12 to eliminate the fractions:

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

Simplify each term:

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

Expand and combine like terms:

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

$17x - 2 = 2x + 10$

Subtract $2x$ from both sides:

$15x - 2 = 10$

Add 2 to both sides:

$15x = 12$

Divide both sides by 15:

$x = \frac{12}{15} = \frac{4}{5}$

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: 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: Perimeters, Numbers

b) 6

[Solution Description]

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

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

Substitute the expressions for length and width:

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

Simplify inside the parentheses:

$2(3w + 5) = 50$

Distribute the 2:

$6w + 10 = 50$

Subtract 10 from both sides:

$6w = 40$

Divide by 6:

$w = \frac{40}{6} = \frac{20}{3} \approx 6.67$

Since the width must be a positive real number, the width is $\frac{20}{3}$ meters.

Key Concept: Numbers, Combination of Currency Notes

d) 8

[Solution Description]

Let the number of \$10 notes be $x$. Then, the number of \$20 notes is $x + 5$. The total amount can be expressed as:

$10x + 20(x + 5) = 200$

Simplify the equation:

$10x + 20x + 100 = 200$

Combine like terms:

$30x + 100 = 200$

Subtract 100 from both sides:

$30x = 100$

Divide by 30:

$x = \frac{100}{30} = \frac{10}{3} \approx 3.33$

Since the number of notes must be an integer, there is no valid solution with this condition.

Key Concept: Numbers

c) 30 and 15

[Solution Description]

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

$x + y = 45$

$x - y = 15$

To solve these equations, add them together:

$2x = 60$

$x = 30$

Substitute $x = 30$ into the first equation:

$30 + y = 45$

$y = 15$

Therefore, the two numbers are 30 and 15.

Key Concept: Perimeter Problem

b) 10

[Solution Description]

Let the width of the rectangle be $x$ cm. Then, the length is $2x$ cm. The perimeter of a rectangle is given by the formula:

$P = 2(l + w)$

Substituting the given values:

$60 = 2(2x + x)$

Simplifying inside the parentheses:

$60 = 2(3x)$

Multiplying:

$60 = 6x$

Dividing both sides by 6:

$x = 10$

Therefore, the width of the rectangle is 10 cm.

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) 30

[Solution Description]

Let the smaller number be $x$. Then, the larger number is $x + 10$. According to the problem, $x + (x + 10) = 50$. Simplifying this equation, we get $2x + 10 = 50$. Subtracting 10 from both sides, we have $2x = 40$. Dividing both sides by 2, we find $x = 20$. Therefore, the larger number is $x + 10 = 20 + 10 = 30$.

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: 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: 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

a) 36

[Solution Description]

Let the current age of the father be $x$ and the son be $y$. According to the problem:

$x + y = 45$ (Equation 1)

Five years ago, their ages were $(x - 5)$ and $(y - 5)$, respectively. The product of their ages at that time:

$(x - 5)(y - 5) = 124$ (Equation 2)

From Equation 1, $y = 45 - x$. Substituting this into Equation 2:

$(x - 5)(40 - x) = 124$

Expanding this equation:

$-x^2 + 45x - 200 = 124$

Bringing all terms to one side:

$-x^2 + 45x - 324 = 0$

Multiplying by $-1$ to simplify:

$x^2 - 45x + 324 = 0$

Solving the quadratic equation using the quadratic formula:

$x = \frac{45 \pm \sqrt{2025 - 1296}}{2}$

$x = \frac{45 \pm \sqrt{729}}{2}$

$x = \frac{45 \pm 27}{2}$

The possible solutions are $x = 36$ or $x = 9$. Since the father cannot be younger than the son, the father's current age is 36 years.

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: Age-related problems

b) 20

[Solution Description]

Let the current age of the son be $x$ and the father be $y$. According to the problem, five years ago:

$y - 5 = 3(x - 5)$

Simplifying this equation:

$y - 5 = 3x - 15$

$y = 3x - 10$ (Equation 1)

After ten years:

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

Simplifying this equation:

$y + 10 = 2x + 20$

$y = 2x + 10$ (Equation 2)

Equating Equation 1 and Equation 2:

$3x - 10 = 2x + 10$

Solving for $x$:

$x = 20$

Thus, the son's current age is 20 years.

Key Concept: Linear Equations in Perimeter Problems

c) Assertion is true, but Reason is false.

[Solution Description]

The assertion states the correct formula for the perimeter of a rectangle, which is $P = 2(l + w)$. However, knowing only the perimeter does not provide enough information to determine the exact values of both length ($l$) and width ($w$). For example, if the perimeter is 20 units, there are multiple combinations of length and width that satisfy this condition (e.g., $l = 5$, $w = 5$ or $l = 6$, $w = 4$). Therefore, the reason is false because the perimeter alone does not uniquely determine the length and width.

Key Concept: Perimeter-based linear equations

d) 20 cm

[Solution Description]

Let the sides of the triangle be $3x$, $4x$, and $5x$ respectively. The perimeter of the triangle is given as 48 cm. So, we write the equation:

$3x + 4x + 5x = 48$

Combine like terms:

$12x = 48$

Divide both sides by 12:

$x = 4$

The longest side is $5x = 5 \times 4 = 20$ 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 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: 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: Currency conversion, Linear equations

c) 8

[Solution Description]

First, convert euros to dollars:

$5 \text{ euros} = 6 \text{ dollars}$

Therefore, 1 euro is equivalent to:

$\frac{6}{5} \text{ dollars}$

Now, convert 10 euros to dollars:

$10 \times \frac{6}{5} = 12 \text{ dollars}$

Next, convert dollars to pounds:

$3 \text{ dollars} = 2 \text{ pounds}$

Therefore, 1 dollar is equivalent to:

$\frac{2}{3} \text{ pounds}$

Finally, convert 12 dollars to pounds:

$12 \times \frac{2}{3} = 8 \text{ pounds}$

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: Combination of Currency Notes

b) 4

[Solution Description]

Let the number of \$50 notes be $x$. Then, the number of \$20 notes is $2x + 5$.

The total amount from \$50 notes is $50x$, and the total amount from \$20 notes is $20(2x + 5)$.

The total amount is given as \$500, so we can write the equation:

$50x + 20(2x + 5) = 500$

Simplifying the equation:

$50x + 40x + 100 = 500$

Combining like terms:

$90x + 100 = 500$

Subtracting 100 from both sides:

$90x = 400$

Dividing both sides by 90:

$x = \frac{400}{90} \approx 4.44$

Since the number of notes must be a whole number, we round to the nearest whole number, which is 4.

Therefore, the person has 4 \$50 notes.

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.

Key Concept: Application of linear equations in perimeter problems

b) 5 meters

[Solution Description]

Let the width of the rectangle be $x$ meters. Then, the length is $2x$ meters. The perimeter of a rectangle is given by:

$2(\text{Length} + \text{Width}) = 30$

Substituting the values:

$2(2x + x) = 30 \Rightarrow 6x = 30 \Rightarrow x = 5$

Hence, the width of the rectangle is 5 meters.

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: Numbers, Ages

a) 12 years, 36 years

[Solution Description]

Let the present age of the son be $x$ years. Then, the present age of the father is $3x$ years.

After 12 years, the son's age will be $(x + 12)$ years, and the father's age will be $(3x + 12)$ years.

According to the problem, $3x + 12 = 2(x + 12)$. Solving this equation:

$3x + 12 = 2x + 24$

Subtract $2x$ from both sides:

$3x - 2x + 12 = 24$

Simplify:

$x + 12 = 24$

Subtract $12$ from both sides:

$x = 12$

Therefore, the present age of the son is $12$ years, and the present age of the father is $36$ years.