College Algebra Study Guide: Linear Equations, Graphs, and Slopes

Q14. Graph the linear function by plotting the x- and y-intercepts. Show both intercepts.

Background

Topic: Linear Functions and Graphing

This question tests your ability to graph a linear equation by finding and plotting its x- and y-intercepts.

Key Terms and Formulas:

• x-intercept: The point where the graph crosses the x-axis ($y = 0$).

• y-intercept: The point where the graph crosses the y-axis ($x = 0$).

• General linear equation: $y = mx + b$

Step-by-Step Guidance

1. Set $y = 0$ in the equation $y = \frac{1}{3}x - 3$ to find the x-intercept.

2. Set $x = 0$ in the equation to find the y-intercept.

3. Plot both intercepts on a coordinate plane.

4. Draw a straight line through the two intercepts to graph the function.

Try solving on your own before revealing the answer!

Q15. Graph the linear equation $4y = 36$.

Background

Topic: Linear Equations and Graphing

This question tests your ability to graph a simple linear equation, specifically a horizontal line.

Key Terms and Formulas:

• Horizontal line: An equation of the form $y = c$.

• Solving for y: $4y = 36 \implies y = \frac{36}{4}$

Step-by-Step Guidance

1. Divide both sides of the equation by 4 to solve for $y$.

2. Recognize that the solution is a horizontal line at $y = c$.

3. Draw the line across the y-axis at the value you found.

Try solving on your own before revealing the answer!

Q17. Find the slope, if possible, of the line passing through the points (9, 5) and (4, 1).

Background

Topic: Slope of a Line

This question tests your ability to calculate the slope between two points.

Key Terms and Formulas:

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

• $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

Step-by-Step Guidance

1. Label the points: $(x_1, y_1) = (9, 5)$ and $(x_2, y_2) = (4, 1)$.

2. Plug the values into the slope formula: $m = \frac{1 - 5}{4 - 9}$.

3. Simplify the numerator and denominator separately.

Try solving on your own before revealing the answer!

Q18. Find the slope of a line perpendicular to the line passing through (8, 4) and (-1, 7).

Background

Topic: Perpendicular Slopes

This question tests your understanding of how to find the slope of a perpendicular line.

Key Terms and Formulas:

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

• Perpendicular slope: If $m$ is the slope of the original line, the perpendicular slope is $-\frac{1}{m}$.

Step-by-Step Guidance

1. Find the slope of the line passing through $(8, 4)$ and $(-1, 7)$ using the slope formula.

2. Calculate the perpendicular slope by taking the negative reciprocal of the slope you found.

Try solving on your own before revealing the answer!

Q19. Find the slope of a line parallel to the line passing through (-3, 5) and (2, 1).

Background

Topic: Parallel Slopes

This question tests your ability to find the slope of a line parallel to another, using two points.

Key Terms and Formulas:

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

• Parallel lines: Parallel lines have the same slope.

Step-by-Step Guidance

1. Label the points: $(x_1, y_1) = (-3, 5)$ and $(x_2, y_2) = (2, 1)$.

2. Plug the values into the slope formula: $m = \frac{1 - 5}{2 - (-3)}$.

3. Simplify the numerator and denominator separately.

Try solving on your own before revealing the answer!

Q20. Write the equation of the line with slope $-\frac{9}{2}$ and y-intercept $-\frac{43}{5}$ in slope-intercept form.

Background

Topic: Slope-Intercept Form

This question tests your ability to write a linear equation given the slope and y-intercept.

Key Terms and Formulas:

• Slope-intercept form: $y = mx + b$

• $m$ is the slope, $b$ is the y-intercept.

Step-by-Step Guidance

1. Identify the slope ($m = -\frac{9}{2}$) and y-intercept ($b = -\frac{43}{5}$).

2. Substitute these values into the slope-intercept form equation.

Try solving on your own before revealing the answer!

Q22. Sketch the graph of the equation $y = -4x + 5$ using any method.

Background

Topic: Graphing Linear Equations

This question tests your ability to graph a linear equation using slope and y-intercept.

Key Terms and Formulas:

• Slope-intercept form: $y = mx + b$

• Slope ($m$): Indicates the steepness and direction of the line.

• Y-intercept ($b$): The point where the line crosses the y-axis.

Step-by-Step Guidance

1. Identify the slope ($m = -4$) and y-intercept ($b = 5$).

2. Plot the y-intercept at $(0, 5)$ on the graph.

3. Use the slope to find another point: from $(0, 5)$, move down 4 units and right 1 unit.

4. Draw a straight line through the points.

Try solving on your own before revealing the answer!

Q23. Find the equation of the line that has the given slope $-3$ and passes through the point $(-5, 3)$. Write the equation in slope-intercept form.

Background

Topic: Point-Slope and Slope-Intercept Form

This question tests your ability to write the equation of a line given a slope and a point.

Key Terms and Formulas:

• Point-slope form: $y - y_1 = m(x - x_1)$

• Slope-intercept form: $y = mx + b$

Step-by-Step Guidance

1. Plug the slope ($m = -3$) and the point $(-5, 3)$ into the point-slope form.

2. Simplify the equation to solve for $y$ and write it in slope-intercept form.

Try solving on your own before revealing the answer!

Q24. Find the equation of a line passing through the points (2, -19) and (5, 37). Write the equation in slope-intercept form.

Background

Topic: Slope-Intercept Form from Two Points

This question tests your ability to find the equation of a line given two points.

Key Terms and Formulas:

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

• Slope-intercept form: $y = mx + b$

Step-by-Step Guidance

1. Label the points: $(x_1, y_1) = (2, -19)$ and $(x_2, y_2) = (5, 37)$.

2. Calculate the slope using the formula.

3. Use one of the points and the slope to solve for $b$ in $y = mx + b$.

Try solving on your own before revealing the answer!

Q25. Complete the ordered pairs for the given linear equation $y = \frac{2}{3}x - 3$, then plot the points and graph the equation by connecting the points.

Background

Topic: Linear Equations and Graphing

This question tests your ability to generate ordered pairs from a linear equation and graph them.

Key Terms and Formulas:

• Linear equation: $y = mx + b$

• Ordered pairs: $(x, y)$ values that satisfy the equation.

Step-by-Step Guidance

1. Choose values for $x$ (such as $x = 0$, $x = 3$, $x = -3$) and substitute them into the equation to find $y$.

2. Write the ordered pairs you found.

3. Plot these points on a coordinate plane.

4. Connect the points with a straight line to graph the equation.

Try solving on your own before revealing the answer!

Final Answer Examples

Each question's final answer would include the completed calculation, graph, or equation as appropriate. For example, for Q19: $m = \frac{1 - 5}{2 - (-3)} = \frac{-4}{5}$, so the slope of the parallel line is $-\frac{4}{5}$.

For graphing questions, the final answer would show the completed graph with intercepts and points clearly marked.