Slope Problems with Solutions: Find Slopes, Parallel & Perpendicular Lines

Practice solving a variety of slope problems, including finding the slopes of lines, determining slopes of parallel and perpendicular lines, and identifying slopes of horizontal and vertical lines. Step-by-step solutions are provided for each problem to help you understand the methods and concepts.

Problem 1: Find the slope of a line

Find the slope of the line that passes through the points (-1, 0) and (3, 8).

Solution:

The slope \( m \) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 0}{3 - (-1)} = 2 \]

Problem 2: Find the slope of a vertical line

Find the slope of the line that passes through the points (2, 0) and (2, 4).

Solution:

The slope \( m \) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{2 - 2} = \frac{4}{0} \] Division by zero is not allowed in mathematics. Therefore the slope of the line defined by the points (2, 0) and (2, 4) is undefined. The line is vertical and perpendicular to the x-axis.

Problem 3: Find the slope of a horizontal line

Find the slope of the line that passes through the points (7, 4) and (-9, 4).

Solution:

The slope \( m \) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 4}{-9 - 7} = 0 \] The line is horizontal and parallel to the x-axis.

Problem 4: Are three points collinear?

Are the points A(2, 3), B(5, 6), and C(0, -2) collinear?

Solution:

Slope of AB: \[ m_{AB} = \frac{6 - 3}{5 - 2} = 1 \] Slope of BC: \[ m_{BC} = \frac{-2 - 6}{0 - 5} = \frac{-8}{-5} = \frac{8}{5} \] The slopes are not equal; therefore, the points are not collinear.

Problem 5: Slope of a line parallel to a given line

What is the slope of the line parallel to the line given by \( -y = -2x + 4 \)?

Solution:

Rewrite the equation in slope-intercept form: \[ y = 2x - 4 \] The slope of the given line is 2. Lines parallel to it also have slope 2.

Problem 6: Slope of a line perpendicular to a given line

What is the slope of the line perpendicular to the line given by \( -2y = -8x + 9\)?

Solution:

Rewrite the equation in slope-intercept form: \[ y = 4x - \frac{9}{2} \] Slope of given line: \( m_1 = 4 \) Slope of perpendicular line: \( m_2 = -\frac{1}{m_1} = -\frac{1}{4} \)

Problem 7: Determine if a triangle is a right triangle

Is the triangle with vertices A(0, -1), B(2, 1), and C(-4, 3) a right triangle?

Solution:

Slope of AB: \[ m_{AB} = \frac{1 - (-1)}{2 - 0} = 1 \] Slope of AC: \[ m_{AC} = \frac{3 - (-1)}{-4 - 0} = -1 \] Since \( m_{AB} \cdot m_{AC} = -1 \), the lines are perpendicular. Therefore, the triangle is a right triangle.

Problem 8: Slope of a line from equation

What is the slope of the line \( -7y + 8x = 9 \)?

Solution:

Rewrite in slope-intercept form: \[ -7y = -8x + 9 \implies y = \frac{8}{7}x - \frac{9}{7} \] The slope is \( \frac{8}{7} \).

Problem 9: Slope of a horizontal line

What is the slope of the line \( y = 9 \)?

Solution:

The line is horizontal and parallel to the x-axis, so its slope is 0.

Problem 10: Slope of a vertical line

What is the slope of the line \( x = -5\)?

Solution:

The line is vertical and perpendicular to the x-axis, so its slope is undefined.