Suppose that the cost of a wedding was a function of the number of guests attending. If you knew the slope of the function and you also knew how much the wedding would cost if 150 guests attended, could you write a linear equation representing this situation? If so, what form of the equation would be easiest to use? In this Concept, you'll learn about the pointslope form of a linear equation so that you can answer questions like these.
Objectives
You will be able to...
 Write the equation of a line in pointslope form given two points
 Write the equation of a line in pointslope form given a point and a slope
 Determine a point and slope from an equation in pointslope form
 Write the equation of a line in pointslope form given the graph of the line
 Graph a line given an equation in pointslope form
Key Vocabulary & Concepts
 If a point is on a line, we say that the contains "contains the point."
Overview
Equations can be written in many forms. Though you are probably more familiar with slopeintercept form (y = mx + b), this page will provide a second way to write an equation of a line: pointslope form.
The equation of the line between any two points
To write an equation in pointslope form, you need two things:
 A point on the line
 The slope of the line
Examples
Objective 1: Write the equation of a line in pointslope form given two points
Example 1: Write an equation for a line containing (9, 3) and (4, 5).
Solution: Begin by finding the slope.
Instead of trying to find
It doesn't matter which point you use.
You could also use the other ordered pair to write the equation:
These equations may look completely different, but if you rewrite the equations in slopeintercept form (by solving each one for
Objective 2: Write the equation of a line in pointslope form given a point and a slope
Example 2: Write the equation of the linear function in pointslope form.
Solution: This function has a slope of 9.8 and contains the ordered pair (5.5, 12.5). Substituting the appropriate values into pointslope form, we get the following:
Replacing
where the last equation is in slopeintercept form.
Objective 3: Determine a point and slope from an equation in pointslope form
Example 3:
Solution: Begin by rewriting the equation to make it pointslope form:
Example 4:
Solution: Begin by rewriting the equation to make it pointslope form:
Objective 4: Write the equation of a line in pointslope form given the graph of the line
If the graph of a line does not have a "nice" yintercept or doesn't show a yintercept, then writing an equation in pointslope form will be easier and more accurate.
Example 5: Given the graph below, write an equation for the line in pointslope form.
Identify any two points on the line. Below, I have highlighted points A (5, 4) and B(3, 3).
Count or calculate the slope between A and B. The slope is
Objective 5: Graph a line given an equation in pointslope form
If you are given an equation in pointslope form, it is not necessary to rewrite it in slopeintercept form in order to graph it. The pointslope form of the equation gives you enough information so you can graph the line.
Example 6: Make a graph of the line given by the equation
Solution: Begin by rewriting the equation to make it pointslope form:
A slope of
Now draw a line through the two points and extend the line in both directions.
Common Errors
 sign errors: In the equation, the number next to y is the opposite of the ycoordinate
 sign errors: In the equation, the number next to x is the opposite of the xcoordinate
 x_{1} and y_{1} switched: In the equation, x_{1} goes next to x, y_{1} goes next to y
Guided Practice
Rewrite
Solution: Use the Distributive Property to simplify the right side of the equation:
Solve for
Practice
Sample explanations for some of the practice problems below are available by viewing the following video. The problems that have video solutions are labeled, along with the time that they are presented in the video.The problem number in the video and the number in the practice set will not always match. (i.e., the first problem from the video might be practice problem #6). However, the answers for all practice problems can be found at the bottom of this page. CK12 Basic Algebra: Linear Equations in PointSlope Form(9:38).
 What is the equation for a line containing the points
(x1,y1) and(x2,y2) in pointslope form?  In what ways is it easier to use pointslope form rather than slopeintercept form?
Objective 1: Write the equation of a line in pointslope form given two points
Write the equation, in point slope form, of each line being described.
 The line contains the points (–2, 3) and (–1, –2).
 The line contains the points (0, 0) and (1, 2).
 The line contains the points (10, 12) and (5, 25). (video solution @4:03)
 The line contains the points (2, 3) and (0, 3).
 The line contains the points (–4, –2) and (8, 12).

f(−7)=5 andf(3)=−4 (video solution @7:40) 
f(6)=0 andf(0)=6 
f(32)=0 andf(77)=25
Objective 2: Write the equation of a line in pointslope form given a point and a slope
Write the equation, in point slope form, of each line being described.
 The slope is
13 ; they− intercept is –4.  The slope is
−110 and contains the point (10, 2). (video solution @0:00)  The slope is –75 and contains the point (0, 125).
 The slope is 10 and contains the point (8, –2).

The line has a slope of
35 and ay− intercept of –3. (video solution @5:27)  The line has a slope of –6 and a
y− intercept of 0.5. 
m=−15 andf(0)=7 
m=−12 and \begin{align*}f(2)=5\end{align*}  \begin{align*}m=3\end{align*} and \begin{align*}f(2)=9\end{align*}
 \begin{align*}m=\frac{9}{5}\end{align*} and \begin{align*}f(0)=32\end{align*}
 \begin{align*}m=25\end{align*} and \begin{align*}f(0)=250\end{align*}
Objective 3: Determine a point and slope from an equation in pointslope form
Use the equation to determine the coordinates of a point on the line and the slope of the line.
 \begin{align*}y5=3(x2)\end{align*}
 \begin{align*}y1=5(x+7)\end{align*}
 \begin{align*}y+1=\frac{1}{6}(x12)\end{align*}
 \begin{align*}y+15=\frac{8}{9}(x+43)\end{align*}
Objective 4: Write the equation of a line in pointslope form given the graph of the line
Write the equation, in point slope form, of the line whose graph is shown.
Objective 5: Graph a line given an equation in pointslope form
Graph each line.
 \begin{align*}y5=3(x2)\end{align*}
 \begin{align*}y8=5(x+7)\end{align*}
 \begin{align*}y+1=\frac{5}{6}(x+7)\end{align*}
Other problems
Rewrite each equation in point slope form.
 \begin{align*}y2=3(x1)\end{align*}
 \begin{align*}y+4=\frac{2}{3}(x+6)\end{align*}
 \begin{align*}0=x+5\end{align*}
 \begin{align*}y=\frac{1}{4}(x24)\end{align*}
Answers to Practice Problems
Answers to all the practice problems above are located in this document (pdf).