ORCCA Point-Slope Form

Section 4.6 Point-Slope Form

In Section 4.5, we learned that a linear equation can be written in slope-intercept form, $y=mx+b\text{.}$ This section covers an alternative that can often be more useful depending on the application: point-slope form.

Subsection 4.6.1 Point-Slope Motivation and Definition

Starting in 1990, the population of the United States has been growing by about $2.865$ million per year. Also, back in 1990, the population was $253$ million. Since the rate of growth has been roughly constant, a linear model is appropriate. Let's try to write an equation to model this.

We consider using slope-intercept form, but we would need to know the $y$-intercept, and nothing in the background tells us that. We'd need to know the population of the United States in the year 0, before there even was a United States.

We could do some side work to calculate the $y$-intercept, but let's try something else. Here are some things we know:

The slope equation is $m=\frac{y_2-y_1}{x_2-x_1}\text{.}$
The slope is $m=2.865$ (million per year).
One point on the line is $(1990,253)\text{,}$ because in 1990, the population was $253$ million.

If we use the generic $(x,y)$ to represent a point somewhere on this line, then the rate of change between $(1990,253)$ and $(x,y)$ has to be $2.865\text{.}$ So

\begin{equation*} \frac{y-253}{x-1990}=2.865\text{.} \end{equation*}

There is good reason¹ to want to isolate $y$ in this equation:

\begin{align*} \frac{y-253}{x-1990}\amp=2.865\\ y-253\amp=2.865\multiplyright{(x-1990)}\amp\amp\text{(could distribute, but not going to)}\\ y\amp=2.865(x-1990)\addright{253} \end{align*}

It will help us to see that $y$ (population) depends on $x$ (whatever year it is).

This is a good place to stop. We have isolated $y\text{,}$ and three meaningful numbers appear in the population: the rate of growth, a certain year, and the population in that year. This is a specific example of point-slope form. Before we look deeper at point-slope form, let's continue reducing the line equation into slope-intercept form.

\begin{align*} y\amp=2.865(x-1990)+253\\ y\amp=2.865x - 5701.35 + 253\\ y\amp=2.865x - 5448.35 \end{align*}

One concern with slope-intercept form is that it uses the $y$-intercept, which might be somewhat meaningless in the context of an application. For example, here we have found that the $y$-intercept is at $(0,-5448.35)\text{,}$ but what practical use is that? It's nonsense to say that in the year 0, the population of the United States was $-5448.35$ million. It doesn't make sense to have a negative population. It doesn't make sense to talk about the United States population before there even was a United States. And it doesn't make sense to use this model for years earlier than 1990 because the background information says clearly that the rate of change we have applies to years 1990 and later.

For all these reasons, we prefer the equation when it was in the form

\begin{equation*} y=2.865(x-1990)\addright{253} \end{equation*}

Definition 4.6.1. Point-Slope Form.

When $x$ and $y$ have a linear relationship where $m$ is the slope and $(x_0,y_0)$ is some specific point that the line passes through, one equation for this relationship is

\begin{equation} y=m\left(x-x_0\right)+y_0\label{hHk}\tag{4.6.1} \end{equation}

and this equation is called the point-slope form of the line. It is called this because the slope and one point on the line are immediately discernible from the numbers in the equation.

Figure 4.6.2.

Remark 4.6.3. Alternative Point-Slope Form.

It is also common to define point-slope form as

\begin{equation} y-y_0=m\left(x-x_0\right)\label{NOt}\tag{4.6.2} \end{equation}

by subtracting $y_0$ from each side. Some exercises may appear using this form.

Checkpoint 4.6.4.

In Checkpoint 4.6.4, the solution explains that each of the following are acceptable equations for the same line:

\begin{align*} y\amp=3(x-3)+2\amp y\amp=3(x-2)-1 \end{align*}

The first uses $(3,2)$ as a point on the line, and the second uses $(2,-1)\text{.}$ Are those two equations really equivalent? Let's distribute and simplify each of them to get slope-intercept form.

\begin{align*} y\amp=3(x-3)+2\amp y\amp=3(x-2)-1\\ y\amp=3x-9+2\amp y\amp=3x-6-1\\ y\amp=3x-7\amp y\amp=3x-7 \end{align*}

So, yes. It didn't matter which point we used to write a point-slope equation. We get different-looking equations that still represent the same line.

Point-slope form is preferable when we know a line's slope and a point on it, but we don't know the $y$-intercept.

Example 4.6.5.

A spa chain has been losing customers at a roughly constant rate since the year 2010. In 2013, it had $2{,}975$ customers; in 2016, it had $2{,}585$ customers. Management estimated that the company will go out of business once its customer base decreases to $1{,}800\text{.}$ If this trend continues, when will the company close?

The given information tells us two points on the line: $(2013,2975)$ and $(2016,2585)\text{.}$ The slope formula will give us the slope. After labeling those two points as $(\overset{x_1}{2013},\overset{y_1}{2975})$ and $(\overset{x_2}{2016},\overset{y_2}{2585})\text{,}$ we have:

\begin{align*} \text{slope}\amp=\frac{y_2-y_1}{x_2-x_1}\\ \amp=\frac{2585-2975}{2016-2013}\\ \amp=\frac{-390}{3}\\ \amp=-130 \end{align*}

And considering units, this means they are losing $130$ customers per year.

Let's note that we could try to make an equation for this line in slope-intercept form, but then we would need to calculate the $y$-intercept, which in context would correspond to the number of customers in year $0\text{.}$ We could do it, but we'd be working with numbers that have no real-world meaning in this context.

For point-slope form, since we calculated the slope, we know at least this much:

\begin{equation*} y=-130(x-x_0)+y_0\text{.} \end{equation*}

Now we can pick one of those two given points, say $(2013,2975)\text{,}$ and get the equation

\begin{equation*} y=-130(x-2013)+2975\text{.} \end{equation*}

Note that all three numbers in this equation have meaning in the context of the spa chain.

We're ready to answer the question about when the chain might go out of business. Substitute $y$ in the equation with $1800$ and solve for $x\text{,}$ and we will get the answer we seek.

\begin{align*} y\amp=-130(x-2013)+2975\\ \substitute{1800}\amp=-130(x-2013)+2975\\ 1800\subtractright{2975}\amp=-130(x-2013)\\ -1175\amp=-130(x-2013)\\ \divideunder{-1175}{-130}\amp=\divideunder{-130(x-2013)}{-130}\\ 9.038\amp\approx x-2013\\ 9.038\addright{2013}\amp\approx x\\ 2022\amp\approx x \end{align*}

And so we find that at this rate, the company is headed toward a collapse in 2022.

Shown is a graph that represents the scenario. Note that to make a graph of $y=-130(x-2013)+2975\text{,}$ we must first find the point $(2013,2975)$ and from there use the slope of $-130$ to draw the line.

Figure 4.6.6. A Graph of $y=-130(x-2013)+2975$

Checkpoint 4.6.7.

Subsection 4.6.2 Using Two Points to Build a Linear Equation

Since two points can determine a line's location, we can calculate a line's equation using just the coordinates from any two points it passes through.

Example 4.6.8.

A line passes through $(-6,0)$ and $(9,-10)\text{.}$ Find this line's equation in both point-slope and slope-intercept form.

Explanation

We will use the slope formula to find the slope first. After labeling those two points as $(\overset{x_1}{-6},\overset{y_1}{0}) \text{ and } (\overset{x_2}{9},\overset{y_2}{-10})\text{,}$ we have:

\begin{align*} \text{slope}\amp=\frac{y_2-y_1}{x_2-x_1}\\ \amp=\frac{-10-0}{9-(-6)}\\ \amp=\frac{-10}{15}\\ \amp=-\frac{2}{3} \end{align*}

The point-slope equation is $y=-\frac{2}{3}(x-x_0)+y_0\text{.}$ Next, we will use $(9,-10)$ and substitute $x_0$ with $9$ and $y_0$ with $-10\text{,}$ and we have:

\begin{align*} y\amp=-\frac{2}{3}(x-x_0)+y_0\\ y\amp=-\frac{2}{3}(x-9)+(-10)\\ y\amp=-\frac{2}{3}(x-9)-10 \end{align*}

Next, we will change the point-slope equation into slope-intercept form:

\begin{align*} y\amp=-\frac{2}{3}(x-9)-10\\ y\amp=-\frac{2}{3}x+6-10\\ y\amp=-\frac{2}{3}x-4 \end{align*}

Remark 4.6.9.

Note that many other resources use the alternate point-slope form to write their equations. Those equations will always be equivalent to those created using our point-slope form. In Example 4.6.8, we found the point-slope form $y=-\frac{2}{3}(x-9)-10\text{.}$ The alternate point-slope form equation ²would have given us $y+10=-\frac{2}{3}(x-9)\text{.}$ If you solve this equation for $y$ and simplify, you should still get $y=-\frac{2}{3}x-4\text{,}$ as we did earlier.

khanacademy.org/math/algebra/two-var-linear-equations/point-slope/a/point-slope-form-review

Checkpoint 4.6.10.

Subsection 4.6.3 More on Point-Slope Form

We can tell a lot about a linear equation now that we have learned both slope-intercept form and point-slope form. For example, we can know that $y=4x+2$ is in slope-intercept form because it looks like $y=mx+b\text{.}$ It will graph as a line with slope $4$ and vertical intercept $(0,2)\text{.}$ Likewise, we know that the equation $y=-5(x-3)+2$ is in point-slope form because it looks like $y=m(x-x_0)+y_0\text{.}$ It will graph as a line that has slope $-5$ and will pass through the point $(3,2)\text{.}$

Example 4.6.11.

For the equations below, state whether they are in slope-intercept form or point-slope form. Then identify the slope of the line and at least one point that the line will pass through.

$\displaystyle y=-3x+2$
$\displaystyle y=9(x+1)-6$
$\displaystyle y=5-x$
$\displaystyle y=-\frac{12}{5}(x-9)+1$

Explanation

The equation $y=-3x+2$ is in slope-intercept form. The slope is $-3$ and the vertical intercept is $(0,2)\text{.}$
The equation $y=9(x+1)-6$ is in point-slope form. The slope is $9$ and the line passes through the point $(-1,-6)\text{.}$
The equation $y=5-x$ is almost in slope-intercept form. If we rearrange the right hand side to be $y=-x+5\text{,}$ we can see that the slope is $-1$ and the vertical intercept is $(0,5)\text{.}$
The equation $y=-\frac{12}{5}(x-9)+1$ is in point-slope form. The slope is $-\frac{12}{5}$ and the line passes through the point $(9,1)\text{.}$

Remark 4.6.12.

Again, we should note that the alternate point-slope form can be used to identify equations. For example, the equation $y+10=-\frac{2}{3}(x-9)$ matches the alternate point-slope form equation ³ with slope $-\frac{2}{3}$ and the line passes through the point $(9,-10)\text{.}$ Note that both coordinates are the opposite of what they appear to be in the equation with this form.

en.wikipedia.org/wiki/Linear_equation#Point–slope_form

Example 4.6.13.

Consider the graph in Figure 4.6.14.

Find three equations that describe the line shown written in point-slope form. Three integer-valued points are shown for convenience.
Determine the slope-intercept form of the equation of this line.

Figure 4.6.14.

Explanation

To write any of the equations representing this line in point-slope form, we must first find the slope of the line and we can use the slope formula to do so. We will arbitrarily choose $(0,30)$ and $(-5,42)$ as the two points. Inputting these points into the slope formula yields:

\begin{align*} m\amp=\frac{y_2-y_1}{x_2-x_1}\\ \amp=\frac{\substitute{42}-\substitute{30}}{\substitute{-5}-\substitute{0}}\\ \amp=\frac{12}{-5}\\ \amp=-\frac{12}{5} \end{align*}

Thus, the slope of the line is $-\frac{12}{5}\text{.}$

Next, we need to write an equation in point-slope form based on each point shown. Using the point $(0,30)\text{,}$ we have:

\begin{equation*} y=-\frac{12}{5}(x-0)+30 \end{equation*}

(This simplifies to $y=-\frac{12}{5}x+30\text{.}$)

The next point is $(20,-18)\text{.}$ Using this point, we can write an equation for this line as:

\begin{equation*} y=-\frac{12}{5}(x-20)-18 \end{equation*}

Finally, we can also use the point $(-5,42)$ to write an equation for this line:

\begin{equation*} y=-\frac{12}{5}(x-(-5))+42 \end{equation*}

which can also be written as:

\begin{equation*} y=-\frac{12}{5}(x+5)+42 \end{equation*}
As $(0,30)$ is the vertical intercept, we can write the equation of this line in slope-intercept form as $y=-\frac{12}{5}x+30\text{.}$ It's important to note that each of the equations that were written in point-slope form simplify to this, making all four equations equivalent.

Exercises 4.6.4 Exercises

Review and Warmup

1.

Evaluate ${-6A+8c}$ for $A = 4$ and $c = -7\text{.}$

2.

Evaluate ${7B-8C}$ for $B = -8$ and $C = 8\text{.}$

3.

Evaluate

\begin{equation*} \displaystyle\frac{y_2 - y_1}{x_2 - x_1} \end{equation*}

for $x_1 = 13\text{,}$ $x_2 = 6\text{,}$ $y_1 = 2\text{,}$ and $y_2 = 4\text{:}$

4.

Evaluate

\begin{equation*} \displaystyle\frac{y_2 - y_1}{x_2 - x_1} \end{equation*}

for $x_1 = 17\text{,}$ $x_2 = -8\text{,}$ $y_1 = 19\text{,}$ and $y_2 = -7\text{:}$

Point-Slope Form

5.

A line’s equation is given in point-slope form:

${y}={5\!\left(x-5\right)+28}$

This line’s slope is .

A point on this line that is apparent from the given equation is .

6.

A line’s equation is given in point-slope form:

${y}={2\!\left(x-1\right)+5}$

This line’s slope is .

A point on this line that is apparent from the given equation is .

7.

A line’s equation is given in point-slope form:

${y}={-2\!\left(x+2\right)+5}$

This line’s slope is .

A point on this line that is apparent from the given equation is .

8.

A line’s equation is given in point-slope form:

${y}={-3\!\left(x+4\right)+7}$

This line’s slope is .

A point on this line that is apparent from the given equation is .

9.

A line’s equation is given in point-slope form:

$\displaystyle{ {y}={\left(-\frac{5}{6}\right)\!\left(x+12\right)+15} }$

This line’s slope is .

A point on this line that is apparent from the given equation is .

10.

A line’s equation is given in point-slope form:

$\displaystyle{ {y}={\frac{6}{5}\!\left(x+10\right) - 8} }$

This line’s slope is .

A point on this line that is apparent from the given equation is .

11.

A line passes through the points $(5,29)$ and $(2,17)\text{.}$ Find this line’s equation in point-slope form.

12.

A line passes through the points $(2,16)$ and $(1,11)\text{.}$ Find this line’s equation in point-slope form.

13.

A line passes through the points $(2,-8)$ and $(4,-18)\text{.}$ Find this line’s equation in point-slope form.

14.

A line passes through the points $(-4,17)$ and $(1,7)\text{.}$ Find this line’s equation in point-slope form.

15.

A line passes through the points $(-10,{-3})$ and $(0,{1})\text{.}$ Find this line’s equation in point-slope form.

16.

A line passes through the points $(6,{8})$ and $(-6,{-10})\text{.}$ Find this line’s equation in point-slope form.

17.

A line’s slope is $3\text{.}$ The line passes through the point $(2,10)\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

18.

A line’s slope is $3\text{.}$ The line passes through the point $(4,15)\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

19.

A line’s slope is $-3\text{.}$ The line passes through the point $(-3,4)\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

20.

A line’s slope is $-3\text{.}$ The line passes through the point $(1,0)\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

21.

A line’s slope is $1\text{.}$ The line passes through the point $(-1,2)\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

22.

A line’s slope is $1\text{.}$ The line passes through the point $(-5,0)\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

23.

A line’s slope is $-1\text{.}$ The line passes through the point $(2,-7)\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

24.

A line’s slope is $-1\text{.}$ The line passes through the point $(-2,-2)\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

25.

A line’s slope is ${{\frac{3}{2}}}\text{.}$ The line passes through the point $(-2,{-1})\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

26.

A line’s slope is ${{\frac{4}{7}}}\text{.}$ The line passes through the point $(-14,{-12})\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

27.

A line’s slope is ${-{\frac{5}{4}}}\text{.}$ The line passes through the point $(12,{-14})\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

28.

A line’s slope is ${-{\frac{6}{7}}}\text{.}$ The line passes through the point $(14,{-14})\text{.}$ Find an equation for this line in both point-slope and slope-intercept form.

An equation for this line in point-slope form is: .

An equation for this line in slope-intercept form is: .

Point-Slope and Slope-Intercept

29.

Change this equation from point-slope form to slope-intercept form.

${y}={4\!\left(x+1\right)-3}$

In slope-intercept form:

30.

Change this equation from point-slope form to slope-intercept form.

${y}={5\!\left(x-3\right)+12}$

In slope-intercept form:

31.

Change this equation from point-slope form to slope-intercept form.

${y}={-2\!\left(x+3\right)+11}$

In slope-intercept form:

32.

Change this equation from point-slope form to slope-intercept form.

${y}={-5\!\left(x-1\right)-4}$

In slope-intercept form:

33.

Change this equation from point-slope form to slope-intercept form.

$\displaystyle{ {y}={{\frac{2}{3}}\!\left(x+3\right)-7} }$

In slope-intercept form:

34.

Change this equation from point-slope form to slope-intercept form.

$\displaystyle{ {y}={{\frac{3}{8}}\!\left(x+16\right)-11} }$

In slope-intercept form:

35.

Change this equation from point-slope form to slope-intercept form.

$\displaystyle{ {y}={-{\frac{4}{9}}\!\left(x+18\right)+11} }$

In slope-intercept form:

36.

Change this equation from point-slope form to slope-intercept form.

$\displaystyle{ {y}={-{\frac{5}{3}}\!\left(x+6\right)+8} }$

In slope-intercept form: