Step-by-Step Guidance for College Algebra Sample Questions

Q1. Solve for $x$ in the equation: $2x = 45$

Background

Topic: Solving Linear Equations

This question tests your ability to solve a simple linear equation for the variable $x$.

Key Terms and Formulas

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

• Isolate the variable: Use inverse operations to get $x$ alone on one side of the equation.

Step-by-Step Guidance

1. Start with the equation: $2x = 45$.

2. To isolate $x$, divide both sides of the equation by $2$.

3. Set up the division: $x = \frac{45}{2}$.

Try solving on your own before revealing the answer!

Final Answer: $x = 22.5$

Dividing both sides by $2$gives$x = 22.5$.

Q2. Solve for $x$ in the equation: $3(x-2) = 15$

Background

Topic: Solving Linear Equations with Parentheses

This question tests your ability to expand and solve a linear equation involving parentheses.

Key Terms and Formulas

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

• Isolate the variable: Get $x$ alone on one side.

Step-by-Step Guidance

1. Start with the equation: $3(x-2) = 15$.

2. Apply the distributive property: $3 \times x - 3 \times 2$.

3. Simplify to get $3x - 6 = 15$.

4. Add $6$to both sides to isolate the $3x$ term.

5. Set up the equation to solve for $x$ by dividing both sides by $3$.

Try solving on your own before revealing the answer!

Final Answer: $x = 7$

After simplifying and isolating $x$, you get $x = 7$.

Q3. Solve for $x$ in the equation: $x^2 - 5x + 6 = 0$

Background

Topic: Solving Quadratic Equations by Factoring

This question tests your ability to factor a quadratic equation and find its solutions.

Key Terms and Formulas

• Quadratic Equation: $ax^2 + bx + c = 0$

• Factoring: Expressing the quadratic as a product of two binomials.

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

Step-by-Step Guidance

1. Start with the equation: $x^2 - 5x + 6 = 0$.

2. Factor the quadratic into two binomials: $(x - a)(x - b) = 0$.

3. Find two numbers that multiply to $6$and add up to$-5$.

4. Write the factored form and set each factor equal to zero.

Try solving on your own before revealing the answer!

Final Answer: $x = 2$ or $x = 3$

The quadratic factors as $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$.

Q4. Simplify the expression: $4(2x-3) - (x+5)$

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: $a(b + c) = ab + ac$

• Like Terms: Terms with the same variable raised to the same power.

Step-by-Step Guidance

1. Apply the distributive property to $4(2x-3)$ to expand the expression.

2. Distribute the negative sign to $(x+5)$.

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

Try solving on your own before revealing the answer!

Final Answer: $7x - 17$

After distributing and combining like terms, the simplified expression is $7x - 17$.

Q5. Solve for $x$ in the equation: $2x + 10 = 0$

Background

Topic: Solving Linear Equations

This question tests your ability to solve a basic linear equation for $x$.

Key Terms and Formulas

• Linear Equation: $ax + b = 0$

• Isolate the variable: Use inverse operations to solve for $x$.

Step-by-Step Guidance

1. Start with the equation: $2x + 10 = 0$.

2. Subtract $10$from both sides to isolate the $2x$ term.

3. Divide both sides by $2$to solve for$x$.

Try solving on your own before revealing the answer!

Final Answer: $x = -5$

Subtracting $10$and dividing by $2$ gives $x = -5$.

Q6. Solve for $x$ in the equation: $x + 1 = 3(x - 2)$

Background

Topic: Solving Linear Equations with Variables on Both Sides

This question tests your ability to solve a linear equation where $x$ appears on both sides.

Key Terms and Formulas

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

• Collect like terms: Move all $x$ terms to one side and constants to the other.

Step-by-Step Guidance

1. Start with the equation: $x + 1 = 3(x - 2)$.

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

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

4. Simplify both sides to isolate $x$.

Try solving on your own before revealing the answer!

Final Answer: $x = 2$

After simplifying and isolating $x$, you find $x = 2$.