College Algebra Final Exam Review: Step-by-Step Guidance

Q1. Evaluate the expression for $a = -5$, $b = 16$, and $c = 7$: $\frac{2c - a^2}{5a + c^2}$

Background

Topic: Evaluating Algebraic Expressions

This question tests your ability to substitute given values into an algebraic expression and simplify the result.

Key Terms and Formulas:

• Substitution: Replacing variables with their given values.

• Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

Step-by-Step Guidance

1. Substitute $a = -5$, $b = 16$, and $c = 7$ into the expression $\frac{2c - a^2}{5a + c^2}$.

2. Calculate $a^2$ and $c^2$ first, since exponents come before multiplication and addition.

3. Compute $2c$ and $5a$.

4. Plug these computed values into the numerator and denominator, but do not simplify to the final value yet.

Try solving on your own before revealing the answer!

Q2. Simplify the expression: $5p + 3(3p + 4) - 3(5p + 3) - 3(2p + 4)$

Background

Topic: Simplifying Algebraic Expressions

This question tests your ability to use the distributive property and combine like terms.

Key Terms and Formulas:

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

• Combining Like Terms: Add or subtract coefficients of the same variable.

Step-by-Step Guidance

1. Apply the distributive property to each term with parentheses: $3(3p + 4)$, $-3(5p + 3)$, and $-3(2p + 4)$.

2. Write out the expanded expression after distributing.

3. Combine all like terms (all $p$ terms together, all constants together).

4. Simplify the expression as much as possible, but stop before the final answer.

Try solving on your own before revealing the answer!

Q3. Solve the equation: $\frac{4x + 3}{2} + 1 = -\frac{4x}{3}$

Background

Topic: Solving Linear Equations with Fractions

This question tests your ability to solve linear equations that involve fractions and variables on both sides.

Key Terms and Formulas:

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

• Clearing Fractions: Multiply both sides by the least common denominator (LCD) to eliminate fractions.

Step-by-Step Guidance

1. Identify the denominators in the equation: 2 and 3.

2. Find the least common denominator (LCD) for 2 and 3.

3. Multiply both sides of the equation by the LCD to clear the fractions.

4. Expand and simplify both sides, then collect like terms, but do not solve for $x$ yet.

Try solving on your own before revealing the answer!

Q4. Solve for $y$: $-0.08y + 0.09(4000 - y) = 0.08y$

Background

Topic: Solving Linear Equations

This question tests your ability to solve a linear equation with variables on both sides and to use the distributive property.

Key Terms and Formulas:

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

• Combining Like Terms: Add or subtract coefficients of the same variable.

Step-by-Step Guidance

1. Apply the distributive property to $0.09(4000 - y)$.

2. Combine all $y$ terms on one side and constants on the other.

3. Simplify the equation to isolate $y$, but do not solve for $y$ yet.

Try solving on your own before revealing the answer!