Imagine you're skateboarding down a ramp or climbing a hill on your favorite hiking trail. Ever wonder how steep it is?
The point-slope form can help you figure that out!
This equation is all about describing change—like how fast you're going, how high you're climbing, or how quickly something increases or decreases over time.
Whether you're tracking your progress in a race, predicting your savings, or even designing a video game, the point-slope form helps you put those changes into a math equation.
In this guide, we'll break it down step by step, explore real-world uses, and give you fun practice problems to try. Let’s get started!
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Before diving into point-slope form, let’s take a moment to review what slope means in mathematics.
A slope is a measure of how steep a line is. It tells us how much a line rises or falls as we move from one point to another. In real life, a slope appears everywhere—from roads and ramps to stairs and rooftops.
Let’s say you’re hiking up a hill. If the hill is very steep, the slope is large. If it’s a gentle incline, the slope is smaller. Similarly, if a road slopes downward, the slope is negative, meaning you're going downhill.
Mathematically, we can calculate the slope using:
\(\displaystyle m = \frac{\text{rise}}{\text{run}}\)
The rise is the change against the y-axis, while the run is the change against the x-axis, so:
\(\displaystyle m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\)
Understanding slope is key to using point-slope form because it tells us how a line behaves and where it will go when plotted on a graph.
The point-slope form of a linear equation is a way to write the equation of a line when you know one point on the line and its slope. It looks like this:
\(\displaystyle y - y_1 = m(x - x_1)\)
Understanding point-slope form is like having a set of instructions to draw a line. If you know where to start and how steep the line is, you can map out the entire equation with ease.
Let’s break down the point-slope form, explain how it works, and see how to visualize it on a graph.
Let's think about how we describe a line mathematically.
A line is made up of an infinite number of points, but we only need two key pieces of information to define it completely:
How steep the line is, or its slope, which we denote with the letter \(\displaystyle m\)
A point the line passes through on the coordinate plane which we’ll mark as \(\displaystyle (x_1, y_1)\)
We already know that slope measures how much the line rises or falls for every unit it moves horizontally. This is written as:
\(\displaystyle m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\)
Normally, to calculate the slope, we need two points, \(\displaystyle (x_1, y_1) \quad \text{and} \quad (x_2, y_2)\).
But in this case, we only have one point—hence point-slope.
Since a line extends infinitely in both directions, we can describe any other point on the line as simply \(\displaystyle (x, y)\) —these represent all the possible coordinates that satisfy the equation of the line.
We can now rewrite our slope equation as:
\(\displaystyle m = \frac{y - y_1}{x - x_1}\)
Notice what we did here:
We replaced \(\displaystyle y_2\) with just \(\displaystyle y\) , because we’re looking for an equation that works for any point on the line.
We replaced \(\displaystyle x_2\) with just \(\displaystyle x\), for the same reason.
Since we already know that slope is rise over run, we can rewrite the equation by multiplying both sides by \(\displaystyle x - x_1\) to eliminate the fraction:
\(\displaystyle m = \frac{y - y_1}{x - x_1}\)
\(\displaystyle m(x - x_1) = (x - x_1) \times \frac{y - y_1}{x - x_1}\)
\(\displaystyle m(x - x_1) = y - y_1\)
There we have it!
Though, the point-slope form is usually written in this sequence:
\(\displaystyle y - y_1 = m(x - x_1)\)
This equation tells us that every point \(\displaystyle (x, y)\) on the line follows this relationship, based on the known point \(\displaystyle (x_1, y_1)\) , and the slope \(\displaystyle m\).
For example, if we know a line passes through the point (2,3), and has a slope of 4, we can write:
\(\displaystyle y - 3 = 4(x - 2)\)
This means that if we start at (2,3) and move up 4 units, for every 1 unit to the right, we’ll stay on the line.
If we were to plot this equation on a graph, we would begin by marking the point (2,3).
Then, using the slope, we could plot additional points by following the pattern up 4, right 1. This creates a straight line, showing how the equation directly relates to the graph.
The point-slope form is a powerful tool for writing the equation of a straight line when you know one point on the line and its slope.
Follow the simple steps below to write an equation using point-slope form:
To write an equation in point-slope form, start by identifying:
A point on the line, or rather, its coordinates \(\displaystyle (x_1, y_1)\)
The slope, \(\displaystyle m\)
For example, if we're given the point (3,5), and a slope of 2, these are the values we'll use.
Use the point-slope formula:
\(\displaystyle y - y_1 = m(x - x_1)\)
Substitute the slop and coordinates \(\displaystyle (x_1, y_1)\):
\(\displaystyle y - 5 = 2(x - 3)\)
This equation represents the line passing through (3,5) with a slope of 2.
Sometimes, we may need to convert the equation to slope-intercept form \(\displaystyle y = mx + b\) to make graphing easier.
Most of the values in the slope-intercept form should be familiar to you by now, like m which represents the slope of the line. The only new value we see here is b which is the y-intercept, or the point where the line crosses the y-axis which means that x = 0.
To convert our point-slope form into slope-intercept form, we first distribute the slope (2) on the right side, multiplying it with each value in brackets:
\(\displaystyle y - 5 = 2(x - 3)\)
\(\displaystyle y - 5 = 2x - 6\)
Next, we add the constant term on the left side to both sides of the equation to isolate the 𝑦. Since we have -5 on the left, we add +5 to both sides to cancel it out.
\(\displaystyle y - 5 + 5 = 2x - 6 + 5\)
After canceling the 5’s on the left, we get our y value:
\(\displaystyle y = 2x - 6 + 5\)
And then we simplify:
\(\displaystyle y = 2x - 1\)
Now, the equation is in slope-intercept form.
Since we know that equation to slope-intercept form is \(\displaystyle y = mx + b\) , we can conclude from the given result \(\displaystyle y = 2x - 1\) that the slope is 2 and the y-intercept is b = -1.
We know that this line intercepts the y-axis at b = -1 and that it’s slope is 2 which means that for every 2 units we go up, the line moves by 1 unit to the side. Let’s graph this:
Now that we understand point-slope form, let’s explore how we can use it to solve equations.
Let’s start with an easy example!
If a line passes through the point (-1,4) and has a slope of -3, what is its point-slope form?
\(\displaystyle x_1 = -1\)
\(\displaystyle y_1 = 4\)
\(\displaystyle m = -3\)
To answer this question, we go back to the formula: \(\displaystyle y - y_1 = m(x - x_1)\) and substitute the values:
\(\displaystyle y - 4 = -3(x - (-1))\)
We know that subtracting a negative value (-1) turns into addition, so the answer is:
\(\displaystyle y - 4 = -3(x + 1)\)
Let’s read a point-slope form to determine our slope and coordinates.
\(\displaystyle y - 3 = 4(x - 7)\)
What does this equation tell us?
Knowing the point-slope form \(\displaystyle y - y_1 = m(x - x_1)\), we can conclude that:
The slope (\(\displaystyle m\)) of this line is 4
Our \(\displaystyle x_1\) coordinate is 7
Our \(\displaystyle y_1\) coordinate is 3
With this information, we can graph the line starting with the point (7,3) and moving up by 4 units, for every 1 unit to the right.
Now let’s find the point-slope form of a line that passes through (1, 3) with a slope of 2, and convert it into slope-intercept: \(\displaystyle y = mx + b\).
Step 1: Identify the Values
We have our \(\displaystyle (x_1, y_1)\) coordinates and our slope m:
\(\displaystyle x_1 = 1\)
\(\displaystyle y_1 = 3\)
\(\displaystyle m = 2\)
Step 2: Write the Point-Slope Equation
We’ll use the point-slope formula:
\(\displaystyle y - y_1 = m(x - x_1)\)
And substitute the values:
\(\displaystyle y - 3 = 2(x - 1)\)
Step 3: Convert into Slope-Intercept Form
If we continue solving, or simplifying, the point-slope equation, we’ll turn it into slope-intercept form \(\displaystyle y = mx + b\) which makes it easier to graph.
First, we distribute 2 to both terms inside the parentheses:
\(\displaystyle y - 3 = 2(x - 1)\)
\(\displaystyle y - 3 = 2x - 2\)
Now we want to isolate the 𝑦. We can do this by adding 3 to both sides:
\(\displaystyle y - 3 + 3 = 2x - 2 + 3\)
\(\displaystyle y = 2x + 1\)
And we have arrived at the slope-intercept form: \(\displaystyle y = 2x + 1\)
Here, the slope m is 2 and b, or the y-intercept, is 1. In other words, our line intercepts the y-axis at y=1.
Point-slope form can help us track patterns and make predictions.
For example, have you ever wondered how many steps you take during a walk?
You start at your house (0 minutes, 0 steps) and walk 50 steps per minute. After 3 minutes, you’ve taken 150 steps.
With point-slope form, you can create an equation to figure out how many steps you’ll take at any time of your walk.
Planning your savings is another great way to see point-slope form in action.
Let’s say you have $10 and decide to save $5 each week. After 4 weeks, you will have $30.
Using point-slope form, you can predict how much money you’ll have after any number of weeks.
Whether you’re tracking steps, saving money, or planning anything that increases at a steady rate, the point-slope form can help!
Now that you know how point-slope form works, try to solve these problems to see if you can apply what you've learned.
If the point-slope equation is \( \displaystyle y - 5 = -4(x - 2) \), what are the \( \displaystyle (x_1, y_1) \) coordinates and the slope.
If a line passes through the point (4,−2) with a slope of 3, what equation represents the line in point-slope form?
Here’s a more challenging one! A line has a slope of \( \displaystyle \frac{2}{5} \)and passes through the point (−3,7). Write its equation in point-slope form and then convert it to slope-intercept form.
When you’re finished, check your results at the bottom of this guide.
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If you’ve given our exercises a go, check your answers below:
Our coordinates and slope are:
\( \displaystyle x_1 = 2 \)
\( \displaystyle y_1 = 5 \)
\( \displaystyle m = -4 \)
\( \displaystyle y - (-2) = 3(x - 4) \)
\( \displaystyle y + 2 = 3(x - 4) \)
Point-slope: \( \displaystyle y - 7 = \frac{2}{5}(x - (-3)) \)
\( \displaystyle y - 7 = \frac{2}{5}(x + 3) \)
Let’s solve the right side by multiplying \( \displaystyle \frac{2}{5} \)with each value in the brackets:
\( \displaystyle y - 7 = \frac{2}{5}x + \frac{2}{5} \times 3 \)
\( \displaystyle y - 7 = \frac{2}{5}x + \frac{6}{5} \)
To isolate the y, we’ll add 7 to both sides:
\( \displaystyle y - 7 + 7 = \frac{2}{5}x + \frac{6}{5} + 7 \)
\( \displaystyle y = \frac{2}{5}x + \frac{6}{5} + 7 \)
To add \( \displaystyle \frac{6}{5} + 7 \), we convert \( \displaystyle 7 \) (which, in fraction form, is \( \displaystyle \frac{7}{1} \) into the equivalent fraction with the denominator \( \displaystyle \frac{x}{5} \):
\( \displaystyle \frac{7 \times 5}{1 \times 5} = \frac{35}{5} \)
Let’s plug that back in
\( \displaystyle y = 25x + 65 + 355 \)
\( \displaystyle y = 25x + 415 \)
Turn \( \displaystyle 415 \) into mixed number \( \displaystyle 815 \)
Slope intercept: \( \displaystyle y = 25x + 815 \)
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