Step-by-Step Guidance for Practice Exam 2 (College Algebra)

Q1. Solve the system of equations:

$3x + 3y = 14$ $5x + 6y = 24$

Background

Topic: Systems of Linear Equations

This question tests your ability to solve a system of two linear equations in two variables using either the substitution or elimination method.

Key Terms and Formulas:

• System of equations: Two or more equations with the same variables.

• Elimination method: Add or subtract equations to eliminate one variable.

• Substitution method: Solve one equation for one variable and substitute into the other.

Step-by-Step Guidance

1. Start by aligning the equations: $3x + 3y = 14$ $5x + 6y = 24$

2. Notice that the coefficients of $y$ in the first equation is 3 and in the second is 6. Multiply the first equation by 2 to match the $y$ coefficients: $2 \times (3x + 3y) = 2 \times 14$ $6x + 6y = 28$

3. Now subtract the second equation from this new equation to eliminate $y$: $(6x + 6y) - (5x + 6y) = 28 - 24$

4. Solve for $x$ in the resulting equation.

Try solving on your own before revealing the answer!

Q2. Solve the system of equations:

$3x + 6y = 30$ $5x + 4y = 32$

Background

Topic: Systems of Linear Equations

This question tests your ability to solve a system of two linear equations in two variables using elimination or substitution.

Key Terms and Formulas:

• System of equations

• Elimination method

• Substitution method

Step-by-Step Guidance

1. Write the equations clearly: $3x + 6y = 30$ $5x + 4y = 32$

2. To eliminate $y$, multiply the second equation by 1.5 so the coefficients of $y$ match: $1.5 \times (5x + 4y) = 1.5 \times 32$ $7.5x + 6y = 48$

3. Subtract the first equation from this new equation to eliminate $y$: $(7.5x + 6y) - (3x + 6y) = 48 - 30$

4. Solve for $x$ in the resulting equation.

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Q3. Solve for $x$:

$\frac{1}{x-2} + \frac{1}{4} = \frac{4}{x^2-4}$

Background

Topic: Rational Equations

This question tests your ability to solve equations involving rational expressions by finding a common denominator.

Key Terms and Formulas:

• Rational equation: An equation involving fractions whose numerators and/or denominators contain a variable.

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

Step-by-Step Guidance

1. Factor denominators where possible. Notice that $x^2-4 = (x-2)(x+2)$.

2. Identify the LCD for all terms: $4(x-2)(x+2)$.

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

4. Simplify each term after multiplying by the LCD.

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Q4. Simplify the complex fraction:

$\frac{3r+2}{r+1} \div \left( \frac{1}{r+1} + \frac{5}{r-1} \right)$

Background

Topic: Complex Fractions

This question tests your ability to simplify complex fractions by finding a common denominator and simplifying the resulting expression.

Key Terms and Formulas:

• Complex fraction: A fraction where the numerator, denominator, or both contain fractions.

• To simplify: Combine the terms in the denominator, then divide the numerator by the denominator (multiply by the reciprocal).

Step-by-Step Guidance

1. Combine the terms in the denominator: $\frac{1}{r+1} + \frac{5}{r-1}$.

2. Find the common denominator for the denominator terms: $(r+1)(r-1)$.

3. Rewrite the denominator as a single fraction.

4. Rewrite the original expression as a division of two fractions, then multiply by the reciprocal of the denominator.

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Q5. Solve for the variable:

$\frac{5}{r-2} - \frac{1}{2} = \frac{3}{r+1}$

Background

Topic: Rational Equations

This question tests your ability to solve equations with rational expressions by finding a common denominator and solving for the variable.

Key Terms and Formulas:

• Rational equation

• Least Common Denominator (LCD)

Step-by-Step Guidance

1. Identify the denominators: $r-2$, $2$, and$r+1$.

2. Find the LCD: $2(r-2)(r+1)$.

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

4. Simplify each term and collect like terms.

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Q6. Simplify the expression:

$\frac{1}{m^2+5m+6} + \frac{2}{m^2+m-6}$

Background

Topic: Adding Rational Expressions

This question tests your ability to add rational expressions with different denominators by factoring and finding a common denominator.

Key Terms and Formulas:

• Factor denominators to find the LCD.

• Add fractions with the same denominator by adding numerators.

Step-by-Step Guidance

1. Factor each denominator: $m^2+5m+6 = (m+2)(m+3)$ $m^2+m-6 = (m+3)(m-2)$

2. Identify the LCD: $(m+2)(m+3)(m-2)$.

3. Rewrite each fraction with the LCD as the denominator.

4. Add the numerators over the common denominator.

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Q7. Simplify the fraction:

$\frac{10xy+15y-4x-6}{10x^2+11x-6}$

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor polynomials in the numerator and denominator and reduce the fraction to lowest terms.

Key Terms and Formulas:

• Factor numerator and denominator completely.

• Cancel any common factors.

Step-by-Step Guidance

1. Factor the numerator by grouping: $10xy+15y-4x-6$.

2. Factor the denominator: $10x^2+11x-6$.

3. Write the expression as a product of factors.

4. Cancel any common factors between numerator and denominator.

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Q8. Graph the equation and find its slope:

$5x - 3y = -12$

Background

Topic: Linear Equations and Slope

This question tests your ability to rewrite a linear equation in slope-intercept form and identify the slope.

Key Terms and Formulas:

• Slope-intercept form: $y = mx + b$, where $m$ is the slope.

• To find the slope, solve for $y$ in terms of $x$.

Step-by-Step Guidance

1. Start with the equation: $5x - 3y = -12$.

2. Solve for $y$ by isolating it on one side of the equation.

3. Rewrite the equation in the form $y = mx + b$ and identify the slope $m$.

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Q9. What is the slope of the line $y = 4$? Sketch the graph of the line.

Background

Topic: Horizontal and Vertical Lines

This question tests your understanding of the slope of horizontal lines and how to graph them.

Key Terms and Formulas:

• Horizontal line: $y = c$ where $c$ is a constant.

• The slope of a horizontal line is $0$.

Step-by-Step Guidance

1. Recognize that $y = 4$ is a horizontal line crossing the $y$-axis at $4$.

2. Recall that the slope of any horizontal line is $0$.

3. Sketch the line by drawing a straight line parallel to the $x$-axis at $y = 4$.

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Q10. Find an equation of the line passing through the points $(2, -5)$ and $(1, 7)$. Write your answer in slope-intercept and standard forms.

Background

Topic: Equations of Lines

This question tests your ability to find the equation of a line given two points, and to write it in both slope-intercept and standard forms.

Key Terms and Formulas:

• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Slope-intercept form: $y = mx + b$

• Standard form: $Ax + By = C$

Step-by-Step Guidance

1. Label the points: $(x_1, y_1) = (2, -5)$ and $(x_2, y_2) = (1, 7)$.

2. Calculate the slope $m$ using the slope formula.

3. Use the point-slope form $y - y_1 = m(x - x_1)$ to write the equation.

4. Convert the equation to slope-intercept form ($y = mx + b$) and then to standard form ($Ax + By = C$).

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Q11. Solve for the variable:

$3x(x+2) - 7 = 2x^2 + 7x + 5$

Background

Topic: Quadratic Equations

This question tests your ability to solve quadratic equations by expanding, combining like terms, and factoring or using the quadratic formula.

Key Terms and Formulas:

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

• Factoring or quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step-by-Step Guidance

1. Expand $3x(x+2)$ to get $3x^2 + 6x$.

2. Rewrite the equation: $3x^2 + 6x - 7 = 2x^2 + 7x + 5$.

3. Move all terms to one side to set the equation to zero.

4. Combine like terms and factor or use the quadratic formula to solve for $x$.

Try solving on your own before revealing the answer!