Step-by-Step Guidance for College Algebra Final Exam Practice

Q1. Solve for x: $8(5x+6) = 11 - (x+7)$

Background

Topic: Linear Equations

This question tests your ability to solve a linear equation in one variable 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 both sides: Expand $8(5x+6)$ and $-(x+7)$.

2. Combine like terms on each side to simplify the equation.

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

4. Factor out $x$ and isolate it by dividing both sides by the coefficient of $x$.

Try solving on your own before revealing the answer!

Q2. Solve for x: $\frac{3}{10}x - 7x + \frac{6}{5} = \frac{3}{5}$

Background

Topic: Linear Equations with Fractions

This problem involves solving a linear equation that contains fractional coefficients. You'll need to combine like terms and clear fractions.

Key Terms and Formulas:

• Least Common Denominator (LCD): The smallest number that is a common denominator for all fractions in the equation.

• Combine like terms: Add or subtract terms with the same variable.

Step-by-Step Guidance

1. Find the least common denominator (LCD) for all fractions in the equation.

2. Multiply both sides of the equation by the LCD to eliminate denominators.

3. Combine like terms and simplify the equation.

4. Isolate $x$ by moving all $x$ terms to one side and constants to the other.

Try solving on your own before revealing the answer!

Q3. Solve for t: $\frac{1}{3}(8t - 4) = \frac{1}{12}(t + 4)$

Background

Topic: Linear Equations with Fractions

This question tests your ability to solve a linear equation with fractional coefficients by clearing denominators and isolating the variable.

Key Terms and Formulas:

• Clear denominators: Multiply both sides by the least common denominator (LCD).

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

Step-by-Step Guidance

1. Identify the LCD for the denominators (3 and 12).

2. Multiply both sides of the equation by the LCD to eliminate fractions.

3. Expand and simplify both sides using the distributive property.

4. Combine like terms and solve for $t$.

Try solving on your own before revealing the answer!

Q4. Solve by the zero-factor property: $w^2 - 13w + 40 = 0$

Background

Topic: Quadratic Equations (Factoring)

This problem tests your ability to solve a quadratic equation by factoring and applying the zero-product property.

Key Terms and Formulas:

• Zero-Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.

• Factoring quadratics: Express $ax^2 + bx + c$ as $(x + p)(x + q)$.

Step-by-Step Guidance

1. Write the quadratic in standard form: $w^2 - 13w + 40 = 0$.

2. Factor the quadratic expression into two binomials.

3. Set each factor equal to zero and solve for $w$.

Try solving on your own before revealing the answer!

Q5. Solve by the zero-factor property: $2y^2 + 5y = -3$

Background

Topic: Quadratic Equations (Factoring)

This question requires you to solve a quadratic equation by first moving all terms to one side, then factoring and applying the zero-product property.

Key Terms and Formulas:

• Zero-Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.

• Factoring quadratics: Express $ax^2 + bx + c$ as $(x + p)(x + q)$.

Step-by-Step Guidance

1. Move all terms to one side to set the equation to zero: $2y^2 + 5y + 3 = 0$.

2. Factor the quadratic expression.

3. Set each factor equal to zero and solve for $y$.

Try solving on your own before revealing the answer!

Q6. Solve by the zero-factor property: $4b^2 - 17b = 15$

Background

Topic: Quadratic Equations (Factoring)

This problem tests your ability to solve a quadratic equation by moving all terms to one side, factoring, and applying the zero-product property.

Key Terms and Formulas:

• Zero-Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.

• Factoring quadratics: Express $ax^2 + bx + c$ as $(x + p)(x + q)$.

Step-by-Step Guidance

1. Move all terms to one side: $4b^2 - 17b - 15 = 0$.

2. Factor the quadratic expression.

3. Set each factor equal to zero and solve for $b$.

Try solving on your own before revealing the answer!