Lesson 9

Equations of Lines

Let’s investigate equations of lines.

Line. Points plotted at -10 comma -6 and 5 comma 2.

The slope of the line in the image is $\frac{8}{15}$ . Explain how you know this is true.

The image shows a line.
1. Write an equation that says the slope between the points $(1,3)$ and $(x,y)$ is 2.
2. Look at this equation: $y-3=2(x-1)$
  How does it relate to the equation you wrote?
Here is an equation for another line: $y-7=\frac12 (x-5)$
1. What point do you know this line passes through?
2. What is the slope of this line?
Next, let’s write a general equation that we can use for any line. Suppose we know a line passes through a particular point $(h,k)$ .
1. Write an equation that says the slope between point $(x,y)$ and $(h,k)$ is $m$ .
2. Look at this equation: $y-k=m(x-h)$ . How does it relate to the equation you wrote?

Write an equation that describes each line.
1. the line passing through point $(\text-2, 8)$ with slope $\frac45$
2. the line passing through point $(0,7)$ with slope $\text-\frac73$
3. the line passing through point $(\frac12, 0)$ with slope -1
4. the line in the image
Using the structure of the equation, what point do you know each line passes through? What’s the line’s slope?
1. $y-5=\frac32 (x+4)$
2. $y+2=5x$
3. $y=\text-2(x-\frac58)$

Are you ready for more?

Another way to describe a line, or other graphs, is to think about the coordinates as changing over time. This is especially helpful if we’re thinking tracing an object’s movement. This example describes the $x$ - and $y$ -coordinates separately, each in terms of time, $t$ .

On the first grid, create a graph of $x=2+5t$ for $\text-2\leq t\leq 7$ with $x$ on the vertical axis and $t$ on the horizontal axis.
On the second grid, create a graph of $y=3-4t$ for $\text-2\leq t\leq 7$ with $y$ on the vertical axis and $t$ on the horizontal axis.
On the third grid, create a graph of the set of points $(2+5t,3-4t)$ for $\text-2\leq t\leq 7$ on the $xy$ -plane.

The line in the image can be defined as the set of points that have a slope of 2 with the point $(3,4)$ . An equation that says point $(x,y)$ has slope 2 with $(3,4)$ is $\frac{y-4}{x-3}=2$ . This equation can be rearranged to look like $y-4=2(x-3)$ .

Line, y intercept = -2., slope of 2. Points plotted at 3 comma 4 and x comma y.

The equation is now in point-slope form, or $y-k=m(x-h)$ , where:

$(x,y)$ is any point on the line
$(h,k)$ is a particular point on the line that we choose to substitute into the equation
$m$ is the slope of the line

Other ways to write the equation of a line include slope-intercept form, $y=mx+b$ , and standard form, $Ax+By=C$ .

To write the equation of a line passing through $(3, 1)$ and $(0,5)$ , start by finding the slope of the line. The slope is $\text-\frac{4}{3}$ because $\frac{5-1}{0-3}=\text-\frac43$ . Substitute this value for $m$ to get $y-k=\text-\frac{4}{3}(x-h)$ . Now we can choose any point on the line to substitute for $(h,k)$ . If we choose $(3, 1)$ , we can write the equation of the line as $y-1=\text-\frac{4}{3}(x-3)$ .

We could also use $(0,5)$ as the point, giving $y-5=\text-\frac{4}{3}(x-0)$ . We can rearrange the equation to see how point-slope and slope-intercept forms relate, getting $y=\text-\frac{4}{3}x+5$ . Notice $(0,5)$ is the $y$ -intercept of the line. The graphs of all 3 of these equations look the same.

point-slope form

The form of an equation for a line with slope $m$ through the point $(h,k)$ . Point-slope form is usually written as $y-k = m(x-h)$ . It can also be written as $y = k + m(x-h)$ .

Lesson 9

9.1: Remembering Slope

9.2: Building an Equation for a Line

9.3: Using Point-Slope Form

Summary

Glossary Entries