Writing an Equation of a Line Parallel to a Given Line

1.14.3 Writing an Equation of a Line Parallel to a Given Line

1.14.3 • Writing an Equation of a Line Parallel to a Given Line

Activity

Find the equation of the line using point-slope form: $m = 2$ , $(3, 1)$

Compare your answer:

$y - 1 = 2 (x - 3)$

Rewrite your equation from Question 1 in slope-intercept form.

Compare your answer:

$y = 2 x - 5$

Find the equation of the line using point-slope form: $m = 2$ , $(- 1, 4)$

Compare your answer:

y − 4 = 2(x + 1)

Rewrite your equation from Question 3 in slope-intercept form.

Compare your answer:

$y = 2 x + 6$

Find the equation of the line using point-slope form: $m = - 4$ , $(2, 1)$

Compare your answer:

$y - 1 = - 4 (x - 2)$

Rewrite your equation from Question 5 in slope-intercept form.

Compare your answer:

$y = - 4 x + 9$

Parallel lines are lines that have the same slope. Because they have the same slope, they will never touch, or intersect.

Find the equation of the line using point-slope form: $m = - 4$ , $(3, - 2)$

Compare your answer:

$y + 2 = - 4 (x - 3)$

Rewrite your equation from Question 7 in slope-intercept form.

Compare your answer:

$y = - 4 x + 10$

Use the graphing tool or technology outside the course.

Access multimedia content

Graph your equations from questions 2 and 4 on the same graph using the Desmos tool.

Compare your answer:

A graph with two lines: a blue line rising from bottom left to top right, intersecting the y-axis at 6; and an orange line rising from bottom right to top left, intersecting the y-axis at -5.

Use the graphing tool or technology outside the course.

Access multimedia content

10.

Graph your equations from questions 6 and 8 on the same graph using the Desmos tool.

Compare your answer:

Graph with two negative-sloping arrows:both cross the y-axis and are parallel to each other.The blue line crosses the y-axis at (0,10). The red line croses the y-axis at (0,9)

11.

What is the same about the equations given in questions 2 and 4?

Compare your answer:

The slopes are the same. They are both 2.

12.

What is the same about the equations given in questions 6 and 8?

Compare your answer:

The slopes are the same. They are both -4.

Self Check

Which equation represents an equation of a line parallel to the line

y = \frac{1}{2} x - 3

that contains the point

(6, 4)

$y = \frac{1}{2}x - 2$
$y = \frac{1}{2}x - 7$
$y = \frac{1}{2}x + 4$
$y = \frac{1}{2}x + 1$

Additional Resources

Find the Line Parallel to a Given Line

In this section, you will use what you know about writing the equation of a line given the slope and a point to write the equation of parallel lines. The graph shows the equation of the line $y = 2 x - 3 .$ The point $P$ $(- 2, 1)$ is also plotted. You can use the fact that parallel lines have the same slope to find a line that is parallel to line $m$ and goes through point $P$ $(- 2, 1)$ .

This figure has a graph of a straight line and a point on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The point (negative 2, 1) is plotted. The line does not go through the point (negative 2, 1).

From the equation, you know the slope of the line is 2. The second line will pass through $(- 2, 1)$ and have slope 2. To graph the line, start at $(- 2, 1)$ and count out the rise and run. The slope is $\frac{r i s e}{r u n} = \frac{2}{1}$ . Count out the rise, 2, and run, 1, and plot the point. You can graph the line as shown. Line $m$ is parallel to line $l$ and goes through point $P$ $(- 2, 1)$ .

This figure has a graph of a two straight lines on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The first line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The points (negative 2, 1) and (negative 1, 3) are plotted. The second line goes through the points (negative 2, 1) and (negative 1, 3).

To find the equation of a line parallel to a line through a given point algebraically, you can use what you know about finding the equation of a line given the slope and a point.

Example

Write an equation of a line parallel to $y = 2 x - 3$ that contains the point $(- 2, 1)$ . Write the equation in slope-intercept form.

Step 1 - Find the slope of the given line.

The line is in slope-intercept form $y = 2 x - 3$

$m = 2$

Step 2 - Find the slope of the parallel line

Parallel lines have the same slope

$m_{1} = 2$

Step 3 - Identify the point

The given point is $(- 2, 1)$

$(x_{1}, y_{1}) = (- 2, 1)$

Step 4 - Substitute values into the point-slope form $y - y_{1} = m (x - x_{1})$

Simplify

$\begin{matrix} y - y_{1} & = & (x - x_{1}) \\ y - 1 & = & (x - (- 2)) \\ y - 1 & = & (x + 2) \\ y - 1 & = & x + 4 \end{matrix}$

Step 5 - Write the equation in slope-intercept form

$y = 2 x + 5$

Look at the graph with the parallel lines shown previously. Does this equation make sense? What is the $y$ -intercept of the line? What is the slope?

You can use this table as a reference for writing an equation of a line parallel to a given line.

Write the equation in slope-intercept form: $y = m x + b$ .

Step 1 - Find the slope of the given line.

Step 2 - Find the slope of the parallel line.

Step 3 - Identify the point.

Step 4 - Substitute the values into the point-slope form: $y - y_{1} = m (x - x_{1})$ .

Step 5 - Simplify.

Step 6 - Write the equation in slope-intercept form: $y = m x + b$ .

Try it

Find the Line Parallel to a Given Line

Write an equation of a line parallel to the line $y = 3 x + 1$ that contains the point $(4, 2)$ . Write the equation in slope-intercept form.

Compare your answer:

Here is how to find a line parallel to a given line and point.

Step 1 - Find the slope of the given line.

$m = 3$

Step 2 - Find the slope of the parallel line.

$m_{1} = 3$

Step 3 - Identify the point.

$(x_{1}, y_{1})$

$(4, 2)$

Step 4 - Substitute the values into the point-slope form: $y - y_{1} = m (x - x_{1})$ .

$\begin{matrix} y - y_{1} & = & m (x - x_{1}) \\ y - 2 & = & 3 (x - 4) \end{matrix}$

Step 5 - Simplify.

$y - 2 = 3 x - 12$

Step 6 - Write the equation in slope-intercept form: $y = m x + b$ .

$y = 3 x - 10$