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

Q1. Solve for x: $2(4x + 3) = 7x + 5$

Background

Topic: Linear Equations

This question tests your ability to solve a linear equation by applying the distributive property and combining like terms.

Key Terms and Formulas

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

• Combining Like Terms: Add or subtract terms with the same variable.

Step-by-Step Guidance

1. Apply the distributive property to expand $2(4x + 3)$: $2 \times 4x$ and $2 \times 3$.

2. Write the expanded equation: $8x + 6 = 7x + 5$.

3. Move all terms containing $x$ to one side and constants to the other side by subtracting $7x$ from both sides.

4. Isolate $x$ by subtracting the constant from both sides.

Try solving on your own before revealing the answer!

Q2. Solve for x: $\frac{4x}{5} - \frac{x}{10} = \frac{x}{5} + 7$

Background

Topic: Linear Equations with Fractions

This question tests your ability to solve linear equations that involve fractions. You may need to find a common denominator to combine terms.

Key Terms and Formulas

• Common Denominator: The least common multiple of the denominators.

• Isolating the Variable: Move all terms with $x$ to one side.

Step-by-Step Guidance

1. Find a common denominator for all fractions (in this case, 10).

2. Rewrite each term with the common denominator: $\frac{4x}{5} = \frac{8x}{10}$ and $\frac{x}{5} = \frac{2x}{10}$.

3. Substitute these into the equation: $\frac{8x}{10} - \frac{x}{10} = \frac{2x}{10} + 7$.

4. Combine like terms and move all $x$ terms to one side.

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Q3. Solve for x: $0.15x - 0.03 = 0.2x + 0.12$

Background

Topic: Linear Equations with Decimals

This question tests your ability to solve linear equations that include decimal coefficients and constants.

Key Terms and Formulas

• Combining Like Terms: Group $x$ terms and constants.

• Isolating the Variable: Move all $x$ terms to one side and constants to the other.

Step-by-Step Guidance

1. Move all $x$ terms to one side by subtracting $0.2x$ from both sides.

2. Move all constants to the other side by adding $0.03$ to both sides.

3. Combine like terms to simplify the equation.

4. Isolate $x$ by dividing both sides by the coefficient of $x$.

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Q4. Solve for x: $5(x - 4) + x = 6(x - 2) - 8$

Background

Topic: Linear Equations with Distribution and Combining Like Terms

This question tests your ability to use the distributive property, combine like terms, and solve for $x$ in a linear equation.

Key Terms and Formulas

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

• Combining Like Terms: Add or subtract terms with the same variable.

Step-by-Step Guidance

1. Apply the distributive property to $5(x - 4)$ and $6(x - 2)$.

2. Expand both sides: $5x - 20 + x = 6x - 12 - 8$.

3. Combine like terms on both sides.

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

Try solving on your own before revealing the answer!