6.6 Finding the Equation of a Line

Finding the Equation of a Line

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How do online retailers know that ‘you may also like’ a particular item based on something you just ordered? How can economists know how a rise in the minimum wage will affect the unemployment rate? How do medical researchers create drugs to target cancer cells? How can traffic engineers predict the effect on your commuting time of an increase or decrease in gas prices? It’s all mathematics.

You are at an exciting point in your mathematical journey as the mathematics you are studying has interesting applications in the real world.

The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. Data is collected and graphed. If the data points appear to form a straight line, an equation of that line can be used to predict the value of one variable based on the value of the other variable.

To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line. In this section we will look at several ways to write the equation of a line. The specific method we use will be determined by what information we are given.

Find an Equation of the Line Given the Slope and y-Intercept

We can easily determine the slope and intercept of a line if the equation was written in slope–intercept form, $y = m x + b .$ Now, we will do the reverse—we will start with the slope and y-intercept and use them to find the equation of the line.

Example 1

Find an equation of a line with slope $−7$ and y-intercept $open parenthesis 0, −1 close parenthesis$ .

Show solution

Since we are given the slope and y-intercept of the line, we can substitute the needed values into the slope–intercept form, $y = m x + b$ .

Name the slope.
Name the y-intercept.
Substitute the values into $y = m x + b .$

Sometimes, the slope and intercept need to be determined from the graph.

Example 2

Find the equation of the line shown.

Show solution

We need to find the slope and y-intercept of the line from the graph so we can substitute the needed values into the slope–intercept form, $y = m x + b$ .

To find the slope, we choose two points on the graph.

The y-intercept is $open parenthesis 0, −4 close parenthesis$ and the graph passes through $open parenthesis 3, −2 close parenthesis$ .

Find the slope by counting the rise and run.	$m = \frac{rise}{run}$
	$m = \frac{2}{3}$
Find the y-intercept.
Substitute the values into $y = m x + b .$
	$The image displays the linear equation y = (2/3)x - 4. The fraction 2/3 is colored red, indicating the slope, while the number 4, part of the y-intercept, is colored blue, with a minus sign preceding it.$

Find an Equation of the Line Given Two Points

When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given.

We have two options so far for finding an equation of a line: slope–intercept or point–slope. Since we will know two points, it will make more sense to use the point–slope form.

But then we need the slope. Can we find the slope with just two points? Yes. Then, once we have the slope, we can use it and one of the given points to find the equation.

Example 4

Find an equation of a line that contains the points $open parenthesis −3, −1 close parenthesis$ and $open parenthesis 2, −2 close parenthesis$ . Write the equation in slope–intercept form.

Show solution

Since we have two points, we will find an equation of the line using the point–slope form. The first step will be to find the slope.

Find the slope of the line through (−3, −1) and (2, −2).	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

	$m = \frac{- 1}{5}$
	$A mathematical equation displays 'm = -1/5' in a clean, white background. The variable 'm' is shown in light blue, the equals sign and negative sign are black, and the fraction '1/5' is in red.$
Choose either point.
Substitute the values into $y - y_{1} = m (x - x_{1}) .$
	$y - (- 2) = - \frac{1}{5} (x - 2)$
	$y + 2 = - \frac{1}{5} x + \frac{2}{5}$
Write in slope–intercept form.	$y = - \frac{1}{5} x - \frac{8}{5}$

Example 5

Find an equation of a line that contains the points $open parenthesis −2, 4 close parenthesis$ and $open parenthesis −2, −3 close parenthesis$ . Write the equation in slope–intercept form when possible. For a vertical line, give its equation in the form x = a and explain why slope–intercept form is not possible.

Show solution

Again, the first step will be to find the slope.

Find the slope of the line through $open parenthesis −2, 4 close parenthesis and open parenthesis −2, −3 close parenthesis$ .	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
	$m = \frac{−3 - 4}{−2 - (−2)}$
	$m = \frac{−7}{0}$
	The slope is undefined.

This tells us it is a vertical line. Both of our points have an x-coordinate of $−2$ . So our equation of the line is $x equals −2$ . Since there is no $y$ , we cannot write it in slope–intercept form.

You may want to sketch a graph using the two given points. Does the graph agree with our conclusion that this is a vertical line?

We have seen that we can use either the slope–intercept form or the point–slope form to find an equation of a line. Which form we use will depend on the information we are given. This is summarized in Table 6.17.

Table 6.17

To Write an Equation of a Line

If given:

Use:

Form:

Slope and y-intercept

slope–intercept

y = m x + b

Slope and a point

point–slope

y - y_{1} = m (x - x_{1})

Two points

point–slope

y - y_{1} = m (x - x_{1})

Practice Makes Perfect

Find an Equation of the Line Given the Slope and y-Intercept

In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.

slope 3 and y-intercept $open parenthesis 0, 5 close parenthesis$

slope 4 and y-intercept $open parenthesis 0, 1 close parenthesis$

Show solution

$y = 4 x + 1$

slope 6 and y-intercept $open parenthesis 0, −4 close parenthesis$

slope 8 and y-intercept $open parenthesis 0, −6 close parenthesis$

Show solution

$y = 8 x - 6$

slope $−1$ and y-intercept $open parenthesis 0, 3 close parenthesis$

slope $−1$ and y-intercept $open parenthesis 0, 7 close parenthesis$

Show solution

$y = − x + 7$

slope $−2$ and y-intercept $open parenthesis 0, −3 close parenthesis$

slope $−3$ and y-intercept $open parenthesis 0, −1 close parenthesis$

Show solution

$y = −3 x - 1$

slope $\frac{3}{5}$ and y-intercept $open parenthesis 0, −1 close parenthesis$

slope $\frac{1}{5}$ and y-intercept $open parenthesis 0, −5 close parenthesis$

Show solution

$y = \frac{1}{5} x - 5$

slope $- \frac{3}{4}$ and y-intercept $open parenthesis 0, −2 close parenthesis$

slope $- \frac{2}{3}$ and y-intercept $open parenthesis 0, −3 close parenthesis$

Show solution

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

slope 0 and y-intercept $open parenthesis 0, −1 close parenthesis$

slope 0 and y-intercept $open parenthesis 0, 2 close parenthesis$

Show solution

$y equals 2$

slope $−3$ and y-intercept $open parenthesis 0, 0 close parenthesis$

slope $−4$ and y-intercept $open parenthesis 0, 0 close parenthesis$

Show solution

$y equals −4 x$

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.

A line passes through the y-intercept (0, -5) and marked point (1, -2); its x-intercept is (5/3, 0). Labeled ticks run from -6 to 6 on both axes.

A descending line passes through (0, 4) and the marked x-intercept (2, 0). Labeled ticks run from -6 to 6 on both axes.

Show solution

$y = −2 x + 4$

An ascending line passes through (0, -3) and the marked x-intercept (6, 0). Labeled ticks run from -8 to 8 on both axes.

An ascending line passes through (0, 2) and marked point (4, 5); its x-intercept is (-8/3, 0). Labeled ticks run from -6 to 6 on both axes.

Show solution

$y = \frac{3}{4} x + 2$

A descending line passes through the y-intercept (0, 3) and marked point (3, -1); its x-intercept is (9/4, 0). Labeled ticks run from -6 to 6 on both axes.

A descending line passes through (0, -1) and marked point (2, -4); its x-intercept is (-2/3, 0). Labeled ticks run from -6 to 6 on both axes.

Show solution

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

A horizontal line y = -2 passes through the marked point (2, -2). Labeled ticks run from -6 to 6 on both axes.

A horizontal line y = 6 passes through the marked point (-3, 6). Labeled ticks run from -8 to 8 on both axes.

Show solution

$y equals 6$

Find an Equation of the Line Given Two Points

In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form when possible. For a vertical line, give its equation in the form x = a and explain why slope–intercept form is not possible.

$open parenthesis 2, 6 close parenthesis$ and $open parenthesis 5, 3 close parenthesis$

$open parenthesis 3, 1 close parenthesis$ and $open parenthesis 2, 5 close parenthesis$

Show solution

$y = −4 x + 13$

$open parenthesis 4, 3 close parenthesis$ and $open parenthesis 8, 1 close parenthesis$

$open parenthesis 2, 7 close parenthesis$ and $open parenthesis 3, 8 close parenthesis$

Show solution

$y = x + 5$

$open parenthesis −3, −4 close parenthesis$ and $open parenthesis 5, minus 2 close parenthesis$

$open parenthesis −5, −3 close parenthesis$ and $open parenthesis 4, −6 close parenthesis$

Show solution

$y = - \frac{1}{3} x - \frac{14}{3}$

$open parenthesis −1, 3 close parenthesis$ and $open parenthesis −6, −7 close parenthesis$

$open parenthesis −2, 8 close parenthesis$ and $open parenthesis −4, −6 close parenthesis$

Show solution

$y = 7 x + 22$

$open parenthesis 6, −4 close parenthesis$ and $open parenthesis −2, 5 close parenthesis$

$open parenthesis 3, −2 close parenthesis$ and $open parenthesis −4, 4 close parenthesis$

Show solution

$y = - \frac{6}{7} x + \frac{4}{7}$

$open parenthesis 0, 4 close parenthesis$ and $open parenthesis 2, −3 close parenthesis$

$open parenthesis 0, −2 close parenthesis$ and $open parenthesis −5, −3 close parenthesis$

Show solution

$y = \frac{1}{5} x - 2$

$open parenthesis 7, 2 close parenthesis$ and $open parenthesis 7, −2 close parenthesis$

$open parenthesis 4, 2 close parenthesis$ and $open parenthesis 4, −3 close parenthesis$

Show solution

$x equals 4$

$open parenthesis −7, −1 close parenthesis$ and $open parenthesis −7, −4 close parenthesis$

$open parenthesis −2, 1 close parenthesis$ and $open parenthesis −2, −4 close parenthesis$

Show solution

$x equals −2$

$open parenthesis 6, 1 close parenthesis$ and $open parenthesis 0, 1 close parenthesis$

$open parenthesis 6, 2 close parenthesis$ and $open parenthesis −3, 2 close parenthesis$

Show solution

$y equals 2$

$open parenthesis 3, −4 close parenthesis$ and $open parenthesis 5, −4 close parenthesis$

$open parenthesis −6, −3 close parenthesis$ and $open parenthesis −1, −3 close parenthesis$

Show solution

$y equals −3$

$open parenthesis 4, 3 close parenthesis$ and $open parenthesis 8, 0 close parenthesis$

$open parenthesis 0, 0 close parenthesis$ and $open parenthesis 1, 4 close parenthesis$

Show solution

$y equals 4 x$

$open parenthesis −2, −3 close parenthesis$ and $open parenthesis −5, −6 close parenthesis$

$open parenthesis −3, 0 close parenthesis$ and $open parenthesis −7, −2 close parenthesis$

Show solution

$y = \frac{1}{2} x + \frac{3}{2}$

$open parenthesis 8, −1 close parenthesis$ and $open parenthesis 8, −5 close parenthesis$

$open parenthesis 3, 5 close parenthesis$ and $open parenthesis −7, 5 close parenthesis$

Show solution

$y equals 5$

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