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