# Lesson 9

Equations of Lines

• Let’s investigate equations of lines.

### 9.1: Remembering Slope

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

### 9.2: Building an Equation for 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?
2. 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?
3. 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?

### 9.3: Using Point-Slope Form

1. 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

2. 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. $$y=\text-2(x-\frac58)$$

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, $$t$$.

1. 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.
2. 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.
3. 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.

### Summary

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

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.

### Glossary Entries

• 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 $$y = k + m(x-h)$$.