Let's Review
When you know the equation of one line, you can use its slope and other given information to write an equation of a line that is parallel or perpendicular to it. The steps for completing this process are shown in the table below.
- Name the slope of the line you were given.
- Determine the slope of the line you need to write an equation for.
- Use the slope and the information in the problem to write an equation in point-slope form.
- Rewrite the point-slope equation in slope-intercept form, if necessary.
Give it a shot!
How well can you write equations for lines that are parallel or perpendicular to a given line? Use the activity below to practice. Read the information on each tab. Write an equation for a line that is either parallel or perpendicular to the line represented by the given equation. Then, check your answer. Remember to pay close attention to which form you are asked to give your answer in.
Write the equation of a line that is parallel to \( y = - 2x + 3 \), if the parallel line passes through the ordered pair \( ( - 4, - 1) \). Express your final answer in slope-intercept form.
\( y = - 2x - 9 \)
If you need help arriving at this answer, click the Solution button.
Step 1: Name the slope of the line you were given. |
The equation \( y = - 2x + 3 \) is expressed in slope-intercept form. The slope is \( m = - 2 \). |
Step 2: Determine the slope of the line you need to write an equation for. |
You need to write an equation of a line that is parallel to \( y = - 2x + 3 \). The new equation must have a slope of \( m = - 2 \) because parallel lines have identical slope values. |
Step 3: Use the slope and the information in the problem to write an equation in point-slope form. |
The slope of the parallel line will be \( m = - 2 \). The ordered pair the parallel line passes through is \( ( - 4, - 1) \). \( y - \left( - 1 \right) = - 2(x - \left( - 4 \right)) \) |
Step 4: Rewrite the point-slope equation in slope-intercept form, if necessary. |
\( y - \left( - 1 \right) = - 2\left( x - \left( - 4 \right) \right) \) \( y + 1 = - 2\left( x + 4 \right) \) \( y + 1 = - 2x - 8 \) \( y = - 2x - 9 \) |
Rania and Portia need to design a second set of racing slides for the city of Geocove's waterpark. The equation that describes their original design is \( y = - \frac{4}{3}x + 20 \).
The new racing slides need to be the same steepness as the original slides but will pass through the ordered pair \( \left( 0,\ 12 \right). \)
Write the equation that describes the second set of racing slides. Express your final equation in slope-intercept form.
\( y = - \frac{4}{3}x + 12 \)
If you need help arriving at this answer, click the Solution button.
Step 1: Name the slope of the line you were given. |
The equation \( y = - \frac{4}{3}x + 20 \) is expressed in slope-intercept form. The slope is \( m = - \frac{4}{3} \). |
Step 2: Determine the slope of the line you need to write an equation for. |
You need to write an equation of a line that has the same steepness as the original. This means the second line needs to have the same slope as the first. The new equation must have a slope of \( m = - \frac{4}{3} \). |
Step 3: Use the slope and the information in the problem to write an equation in point-slope form. |
The slope of the parallel line will be \( m = - \frac{4}{3} \). The ordered pair the parallel line passes through is \( (0,\ 12) \). \( y - 12 = - \frac{4}{3}(x - 0) \) |
Step 4: Rewrite the point-slope equation in slope-intercept form, if necessary. |
\( y - 12 = - \frac{4}{3}\left( x - 0 \right) \) \( y - 12 = - \frac{4}{3}x \) \( y = - \frac{4}{3}x + 12 \) Note: You could also recognize the given point, \( \left( 0,\ 12 \right) \), is the \( y \)-intercept of the parallel line. |
Write the equation of a line that is perpendicular to \( y + 1 = \frac{4}{9}(x - 18) \) and passes through the ordered pair \( (3,\ 9) \). Express your final answer in point-slope form.
\( y - 9 = - \frac{9}{4}(x - 3) \)
If you need help arriving at this answer, click the Solution button.
Step 1: Name the slope of the line you were given. |
The equation \( y + 1 = \frac{4}{9}(x - 18) \) is expressed in point-slope form. The slope is \( m = \frac{4}{9} \). |
Step 2: Determine the slope of the line you need to write an equation for. |
You need to write an equation of a line that is perpendicular to this one. Non-vertical perpendicular lines have opposite reciprocal slopes. The new equation must have a slope of \( m = - \frac{9}{4} \). |
Step 3: Use the slope and the information in the problem to write an equation in point-slope form. |
The slope of the perpendicular line will be \( m = - \frac{9}{4} \). The ordered pair the perpendicular line passes through is \( (3,\ 9) \). \( y - 9 = - \frac{9}{4}(x - 3) \) |
Step 4: Rewrite the point-slope equation in slope-intercept form, if necessary. |
The instructions indicate to express the final equation in point-slope form. No rewriting is necessary. |
A concrete company is pouring some of the sidewalks that guests will use in the Geocove waterpark.
If the equation that describes one of the sidewalks is \( y - 4 = - \frac{5}{3}(x - 3) \), write an equation the describes a perpendicular sidewalk that passes through the ordered pair \( \left( - 16,\ 14 \right). \) Write your final answer in slope-intercept form.
\( y = \frac{3}{5}x + 23.6 \)
If you need help arriving at this answer, click the Solution button.
Step 1: Name the slope of the line you were given. |
The slope of the equation \( y - 4 = - \frac{5}{3}(x - 3) \) is \( m = - \frac{5}{3} \). |
Step 2: Determine the slope of the line you need to write an equation for. |
Non-vertical perpendicular lines have opposite reciprocal slopes. The new equation must have a slope of \( m = \frac{3}{5} \). |
Step 3: Use the slope and the information in the problem to write an equation in point-slope form. |
The slope of the perpendicular line will be \( m = \frac{3}{5} \). The ordered pair the perpendicular line passes through is \( ( - 16,\ 14) \). \( y - 14 = \frac{3}{5}(x - ( - 16)) \) |
Step 4: Rewrite the point-slope equation in slope-intercept form, if necessary. |
\( y - 14 = \frac{3}{5}\left( x - \left( - 16 \right) \right) \) \( y - 14 = \frac{3}{5}\left( x + 16 \right) \) \( y - 14 = \frac{3}{5}x + 9.6 \) \( y = \frac{3}{5}x + 23.6 \) |
Write the equation of a line that is parallel to the \( x \)-axis, if the parallel line passes through the ordered pair \( (0,\ 2) \). Express your final answer in point-slope form.
\( y - 2 = 0(x - 0) \)
If you need help arriving at this answer, click the Solution button.
Step 1: Name the slope of the line you were given. |
The equation of the \( x \)-axis is \( y = 0 \). The slope of this line is \( m = 0 \). |
Step 2: Determine the slope of the line you need to write an equation for. |
Parallel lines have identical slope. The new equation must have a slope of \( m = 0 \). |
Step 3: Use the slope and the information in the problem to write an equation in point-slope form. |
The slope of the parallel line will be \( m = 0 \). The ordered pair the parallel line passes through is \( (0,\ 2) \). \( y - 2 = 0(x - 0) \) |
Step 4: Rewrite the point-slope equation in slope-intercept form, if necessary. |
The instructions indicate to express the final equation in point-slope form. No rewriting is necessary. |
