How to write an equation in point slope form

What is point-slope form?

In this post, you will learn how to determine the point-slope form of a line given two points or given a point and a slope. You will also see how to graph and how to determine x and y intercepts using point-slope form.

Have you ever played with a ball of play-dough? The feeling of taking what was once a meaningless pile and transforming it into your own beautiful creation can bring such satisfaction! The same piece of play-dough and be rearranged to create amazing new experiences.

Likewise, in mathematics, we can take together pieces of information that may feel meaningless and create something beautiful and useful, like an equation or a graph. While the information stays the same, but the way we use it changes, so we express it in a new way. Let’s dive in!

• What is point-slope form?
• Where does the form come from?
• Find point-slope form given two points (example)
• Find point-slope form given slope and a point (example)
• How to graph point-slope form (example)
• Intercepts from point-slope form
• Other forms of equations
• Summary: Point-Slope Form

What is point-slope form?

Below is the equation for point-slope form:

Point-Slope Form

y-y_1=m(x-x_1)

The values of x_1 and y_1 come from a point the line goes through, (x_1,y_1). The value of m is the slope of the line.

Where does the form come from?

Point-slope form of a line is determined by the slope of the line and any point that exists on the line.

The purpose of the form is to describe the equation of the entire line when given a point on the line and the slope. For example, in calculus point-slope form can describe the line tangent to a function at a given x-value.

We can derive the point-slope equation from the slope formula:

m = \dfrac{y_2 - y_1}{x_2 - x_1}

…where (x_1,y_1) and (x_2, y_2) are points on the line. Multiplying both sides by (x_2 - x_1):

y_2 - y_1 = m(x_2 - x_1)

Find point-slope form given two points (example)

You may need to know how to find an equation with two points.

Let’s determine point-slope form using the line that goes through the points (8,-3) and (-2,6). To determine point-slope from using two given points, we must first determine the slope of the line. The slope of a line is determined using:

\dfrac{y_2-y_1}{x_2-x_1}

First, let’s label our points.

Now, let us substitute the correct values into the slope formula.

\dfrac{y_2-y_1}{x_2-x_1}

\dfrac{6-(-3)}{(-2)-8}

\dfrac{9}{-10}

\dfrac{-9}{10}

Therefore, the slope of the line, m is \frac{-9}{10}. 

Now, we can substitute the values for x_1, y_1 and m into the point-slope form formula. We will use the same values for x_1 and y_1 that we used for slope.

y-y_1=m(x-x_1)

y-(-3)=(\dfrac{-9}{10})(x-8)

y+3=\dfrac{-9}{10}(x-8)

Therefore, the equation of the line going through the points (8,-3) and (-2,6) in point-slope form is y+3=\frac{-9}{10}(x-8). This is how to determine point-slope form with two points.

Here’s a quick video demonstration of writing an equation given two points:

Find point-slope form given slope and a point (example)

You may need to find point-slope form using a slope and a point. For instance, if you may need to determine the equation of a line going through the point (-6, 4) with a slope of 11 using point-slope form.

Let’s begin by labeling the point:

Now we know the value of y_1 is 4 and the value of x_1 is -6. We also know the value of m is 11 because m represents slope.

Let’s substitute the values into the formula!

y-y_1=m(x-x_1)

y-4=11(x-(-6))

y-4=11(x+6)

We have determined the point-slope form equation of the line with a slope of 11 that goes through the point (-6,4) is y-4=11(x+6).

For the visual learners, here’s a quick video example of how to write a point-slope equation given a specific point and slope:

How to graph point-slope form (example)

When given the point-slope form of a line, we first must identify the point and the slope in order to create a graph of the line. For example, let us graph y-3=2(x+1).

We’ll start with the form:

y-y_1=m(x-x_1)

We can see that our equation, y-3=2(x+1) has the value of 2 in the place of m. This means the slope of the line is 2.

Additionally, we see that value 3 is in the place of y_1. To determine x_1, we want to rewrite the equation to match point-slope form. Notice the values of x_1 and y_1 are both subtracted.

y-3=2(x+1)

y-3=2(x-(-1))

Now, we can more clearly see that the value of x_1 must be -1.

Because the value of x_1 is -1 and the value of y_1 is 3, we can plot the point (-1,3) on the graph:

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We know the line y-3=2(x+1) goes through the point (-1,3).

After plotting the point, we can determine another point using the slope. The slope of the line is 2. We can rewrite the slope as \frac{2}{1} meaning a rise of 2 and a run of 1. We can begin at (-1,3) and move up 2 units and to the right 1 unit. This lands us at the point (0,5):

Click image to expand

Finally, we can connect the points to create the line:

Click image to expand

Here’s a helpful video on graphing equations:

Intercepts from point-slope form

To begin, let’s determine x and y intercepts using point-slope form. We will begin with an equation:

y+2=\frac{1}{2}(x-4)

To determine the x-intercept, we must set the value of y equal to zero. Remember, when the line crosses the x-axis, the value of y is always zero.

y+2=\frac{1}{2}(x-4)

0+2=\frac{1}{2}(x-4)

Now, we simply solve for the value of x.

0+2=\frac{1}{2}(x-4)

2=\frac{1}{2}x-2

4=\frac{1}{2}x

8=x

Therefore, when the value of y is 0, the value of x is 8. This means the line y+2=\frac{1}{2}(x-4) has an x-intercept at 8. 

Likewise, to determine the y-intercept, we must set the value of x to zero. Remember, when the line crosses the y-axis, the value of x is always zero.

y+2=\frac{1}{2}(x-4)

y+2=\frac{1}{2}(0-4)

y+2=\frac{1}{2}(-4)c

y+2=-2

y=-4

Therefore, when the value of x is 0, the value of y is -4. This means the line y+2=\frac{1}{2}(x-4) has a y-intercept at -4. 

We have found the x and y-intercepts using the point-slope form of a line, y+2=\frac{1}{2}(x-4).

Click here for even more examples of point-slope equation problems.

Other forms of equations

Linear equations can also be written in slope-intercept form. This form allows you to easily identify the slope and the y-intercept of a line. To learn more, read our review post on slope-intercept form.

Slope-Intercept Form: y=mx+b

An equation can also be written in standard form. This form can be very useful to solve systems of equations. To learn more, read our review article on standard form of linear equations.

Standard From: ax+by=c

Summary: Point-Slope Form

• We know how to determine the intercepts using point-slope form.
• We demonstrated how to graph using point-slope form.
• Whether we have two points or a point and a slope, we can create the equation of a line using point-slope form.
• Being able to readily switch between different forms of linear equations can make solving complex problems easier and more enjoyable!

Click here to explore more helpful Albert Algebra 1 review guides.

How do you write an equation in point slope?