College Algebra Exam 1 Study Guide – Step-by-Step Guidance

Q1. Simplify 3x + 5x.

Background

Topic: Combining Like Terms

This question tests your understanding of how to combine like terms in algebraic expressions, a foundational skill in algebra.

Key Terms and Formulas:

• Like Terms: Terms that have the same variable raised to the same power.

• Combining Like Terms: Add or subtract the coefficients of like terms.

Step-by-Step Guidance

1. Identify the like terms in the expression. Here, both terms have the variable $x$.

2. Add the coefficients of $x$: $3x + 5x$ becomes $(3 + 5)x$.

Try solving on your own before revealing the answer!

Final Answer: 8x

We combined the coefficients of $x$ to get $8x$.

Q2. Simplify 7 - 3 + 2.

Background

Topic: Order of Operations and Arithmetic

This question checks your ability to perform basic arithmetic operations in the correct order.

Key Terms and Formulas:

• Order of Operations: For addition and subtraction, work from left to right.

Step-by-Step Guidance

1. Start with the first two numbers: $7 - 3$.

2. Take the result and add $2$ to it.

Try solving on your own before revealing the answer!

Final Answer: 6

First, $7 - 3 = 4$, then $4 + 2 = 6$.

Q3. Simplify $4(2x - 3) - 5(x + 1)$.

Background

Topic: Distributive Property and Combining Like Terms

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

Key Terms and Formulas:

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

• Like Terms: Terms with the same variable and exponent.

Step-by-Step Guidance

1. Apply the distributive property to both terms: $4(2x - 3)$ and $-5(x + 1)$.

2. Expand: $4 \times 2x$, $4 \times (-3)$, $-5 \times x$, $-5 \times 1$.

3. Combine like terms (the $x$ terms and the constants).

Try solving on your own before revealing the answer!

Final Answer: $3x - 17$

After distributing and combining like terms, you get $3x - 17$.

Q4. Simplify $5a - 3(2a - 4)$.

Background

Topic: Distributive Property and Combining Like Terms

This question tests your ability to distribute and combine like terms in algebraic expressions.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Distribute $-3$ to both $2a$ and $-4$ inside the parentheses.

2. Write out the expanded expression.

3. Combine like terms (the $a$ terms and the constants).

Try solving on your own before revealing the answer!

Final Answer: $-a + 12$

After distributing and combining like terms, you get $-a + 12$.

Q5. Simplify $\frac{3x - 2}{4} - \frac{x + 6}{2}$.

Background

Topic: Operations with Rational Expressions

This question tests your ability to subtract fractions with algebraic numerators by finding a common denominator.

Key Terms and Formulas:

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

• Subtracting Fractions: $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$ (if denominators are different).

Step-by-Step Guidance

1. Find a common denominator for $4$and $2$ (which is $4$).

2. Rewrite $\frac{x + 6}{2}$ with denominator $4$.

3. Subtract the numerators over the common denominator.

4. Simplify the numerator by combining like terms.

Try solving on your own before revealing the answer!

Final Answer: $\frac{x - 14}{4}$

After finding a common denominator and simplifying, you get $\frac{x - 14}{4}$.

Q6. Simplify $(2x - 3y) - (4x + y) + 5(3y - x)$.

Background

Topic: Distributive Property and Combining Like Terms

This question tests your ability to distribute, combine like terms, and handle subtraction of algebraic expressions.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Distribute the $5$to both $3y$ and $-x$ in $5(3y - x)$.

2. Expand all terms and rewrite the expression without parentheses.

3. Combine like terms for $x$ and $y$.

Try solving on your own before revealing the answer!

Final Answer: $-3x + 13y$

After distributing and combining like terms, you get $-3x + 13y$.