Step-by-Step Guidance for Solving Linear Equations (Intermediate Algebra)

Q1. Solve for x: 3(x + 2) = 15

Background

Topic: Solving Linear Equations

This question tests your ability to solve a basic linear equation by applying the distributive property and isolating the variable.

Key Terms and Formulas

• Distributive Property: $a(b + c) = ab + ac$

• Linear Equation: An equation of the form $ax + b = c$

Step-by-Step Guidance

1. Apply the distributive property to expand $3(x + 2)$.

2. Write the expanded equation: $3x + 6 = 15$.

3. Subtract 6 from both sides to isolate the term with $x$.

4. Set up the equation to solve for $x$.

Try solving on your own before revealing the answer!

Final Answer: x = 3

Subtracting 6 from both sides gives $3x = 9$, then dividing both sides by 3 yields $x = 3$.

Q2. Solve for x: -3(x - 12) = 45

Background

Topic: Solving Linear Equations with Negative Coefficients

This question tests your ability to solve a linear equation involving a negative coefficient and the distributive property.

Key Terms and Formulas

• Distributive Property: $a(b - c) = ab - ac$

• Linear Equation: An equation of the form $ax + b = c$

Step-by-Step Guidance

1. Expand $-3(x - 12)$ using the distributive property.

2. Write the expanded equation: $-3x + 36 = 45$.

3. Subtract 36 from both sides to isolate the term with $x$.

4. Set up the equation to solve for $x$.

Try solving on your own before revealing the answer!

Final Answer: x = -3

Subtracting 36 from both sides gives $-3x = 9$, then dividing both sides by -3 yields $x = -3$.

Q3. Solve for x: 2 - 4(3 + 4x) = 54

Background

Topic: Solving Linear Equations with Multiple Steps

This question tests your ability to use the distributive property and combine like terms to solve for $x$.

Key Terms and Formulas

• Distributive Property: $a(b + c) = ab + ac$

• Combining Like Terms: Grouping terms with the same variable

Step-by-Step Guidance

1. Expand $-4(3 + 4x)$ using the distributive property.

2. Write the expanded equation: $2 - 12 - 16x = 54$.

3. Combine like terms on the left side.

4. Set up the equation to solve for $x$.

Try solving on your own before revealing the answer!

Final Answer: x = -4

Combining terms gives $-10 - 16x = 54$, then isolating $x$ yields $x = -4$.

Q4. Solve for x: 5 - (1 - 7x) + 6x = 251

Background

Topic: Solving Linear Equations with Parentheses

This question tests your ability to simplify expressions with parentheses and solve for $x$.

Key Terms and Formulas

• Distributive Property: $a(b + c) = ab + ac$

• Combining Like Terms: Grouping terms with the same variable

Step-by-Step Guidance

1. Expand $-(1 - 7x)$ to $-1 + 7x$.

2. Combine all terms: $5 - 1 + 7x + 6x = 251$.

3. Combine like terms to simplify the equation.

4. Set up the equation to solve for $x$.

Try solving on your own before revealing the answer!

Final Answer: x = 18

Combining terms gives $4 + 13x = 251$, then isolating $x$ yields $x = 18$.

Q5. Solve for x: -3 + 5x = 4x - 10

Background

Topic: Solving Linear Equations with Variables on Both Sides

This question tests your ability to solve a linear equation where variables appear on both sides of the equation.

Key Terms and Formulas

• Linear Equation: An equation of the form $ax + b = cx + d$

• Combining Like Terms: Move all terms with $x$ to one side and constants to the other.

Step-by-Step Guidance

1. Subtract $4x$ from both sides to get all $x$ terms on one side.

2. Combine like terms: $5x - 4x$.

3. Move constants to the other side.

4. Set up the equation to solve for $x$.

Try solving on your own before revealing the answer!

Final Answer: x = -7

Combining terms gives $x = -7$ after isolating the variable.