How to Find the Slope-Intercept Form of an Equation

Finding the slope-intercept form of an equation is pretty straightforward — you can do this by rearranging a linear equation in a different form, looking at a graph, or using the coordinates of two points belonging to a line. Being able to write an equation in this form is crucial, as it makes determining the slope 🇺🇸 and y-intercept 🇺🇸 quick and easy, and simplifies graphing a line equation.

In this article, you will learn what the slope-intercept form is and how to find the slope-intercept form of an equation in three different ways. We will show you some slope-intercept form examples and give you all the necessary explanations so that you never have to struggle again!

In the meantime, if you want to know more about slopes, check out our dedicated article, Positive Slope vs. Negative Slope!

The slope-intercept form is one of many ways to write a linear equation. This formula embeds the slope and y-intercept of a line, allowing us to identify these parameters at a glance.

The slope-intercept form gives us the slope of a line at a glance, making it quicker to compare the slopes of two or more lines. It's also the most popular form for graphing, because it provides the coordinates of a point belonging to the line without calculating anything.

You can find the slope-intercept form in several ways, depending on the resources you're given. Here, we will explain how to find the equation in its slope-intercept form in three different ways: rearranging a linear equation in standard form (or any other form), identifying it from a graph, and using two points belonging to the line.

Case 1: Converting from standard form

You can write an equation in the slope-intercept form by rearranging a different linear equation, for instance, in the standard form: $Ax + By = C$. Here is what you have to do:

1. Move the x-term to the right-hand side:

   $By = -Ax+C$

2. Isolate $y$ on the left-hand side:

   $\large \frac{By}{B}=\frac{-Ax+C}{B}$

   $y=\frac{-A}{B}x+\frac{C}{B}$

3. You will obtain an equation in the slope-intercept form:

   $y=mx+b$

   where:

   • $m=\frac{-A}{B}$
   • $b=\frac{C}{B}$

Example 1.1

Let's see how to find the slope-intercept form of $3x+2y=6$:

$3x+2y=6$

$2y=-3x+6$

$\large \frac{2y}{2}=\frac{-3x+6}{2}$

$y=-\frac{3}{2}x+3$

And there you have it! Your slope-intercept form of this linear equation is $y=-\frac{3}{2}x+3$. Its slope is equal to $m=-\frac{3}{2}$, and the y-intercept is $Y(0, 3)$.

Example 1.2

You can also rearrange any other type of linear equation until you have isolated $y$ on the left-hand side. For example:

$-2(x+4)+5y=9$

What even is this equation? Don't worry, you can easily convert it into the slope-intercept form. Remember—leave $y$ alone on the left side:

$-2(x+4)+5y=9$

$-2x-8+5y=9$

$5y=2x+9+8$

$\large \frac{5y}{5}=\frac{2x+17}{5}$

$y=\frac{2}{5}x+\frac{17}{5}$

$y=\frac{2}{5}x+3\frac{2}{5}$

Easy, right? Now try to convert the following linear equations yourself:

• $\frac{x}{4}+\frac{y}{3}=2$
• $0=6x-3y+2$
• $\frac{8}{3}x-\frac{1}{2}y=7$

Case 2: Deriving from a graph

This method will not work in more advanced instances, but it's an easy way to write the slope-intercept form of a simple line graph quickly. Take a look at the following graph:

We will first have to identify the slope of this line, which tells us how many units the line rises or falls for every unit that it goes to the right. In this case, we can easily determine that the slope is equal to $2$, so let's insert this into our slope-intercept form equation:

$y=2x+b$

Now we have to find $b$, that is, the y-coordinate of the y-intercept, $Y(0, b)$. By looking at the graph, we can determine that the line crosses the y-axis at $(0, -1)$, hence $b=-1$. Let's put it all together:

$y=2x-1$

This is your equation in the slope-intercept form. But remember that lines don't always cross the grids at nice, round intervals, so you won't always be able to write the equation using this method.

Case 3: Using two points

If you don't know the equation of a line, but you know two points belonging to the line, you can easily find the slope-intercept form using the coordinates of said points. Just follow these steps:

1. Write down the coordinates of your two points: $A(x_1, y_1)$ and $B(x_2, y_2)$.

2. Calculate the slope using the rise over run formula 🇺🇸:

   $\large m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

3. Substitute your slope and the coordinates of one of the two points in the point-slope form: $y-y_1=m(x-x_1)$.

4. Finally, rearrange the equation so that $y$ is isolated on the left side, and the right side consists of one x-term and one constant.

This is a little bit more complicated, so let's take a look at an example:

Example 3.1

Given $A(2,-1)$ and $B(6,13)$, let's write an equation in the slope-intercept form.

First, calculate the slope:

$\large m=\frac{y_2-y_1}{x_2-x_1}=\frac{13-(-1)}{6-2}=\frac{14}{4}=\frac{7}{2}$

Now, insert the slope and the coordinates of $A$ into the point-slope form:

$y-y_1=m(x-x_1)$

$y-(-1)=\frac{7}{2}(x-2)$

Then, rearrange the equation until you obtain a slope-intercept form:

$y+1=\frac{7}{2}x-\frac{7}{2}\times 2$

$y=\frac{7}{2}x-7-1$

And finally...

$y=\frac{7}{2}x-8$

As you can see, it wouldn't have been easy to determine the slope just by looking at the graph, but now, thanks to the slope-intercept form of our equation, we know that the slope is equal to $\frac{7}{2}$, and that the line intercepts the y-axis at $Y(0,-8)$.

The slope-intercept form of a linear equation is a crucial element of algebra — it makes it easy to identify the slope and y-intercept of a line and is very useful for graphing. Now you know how to express a line equation as y = mx + b every time, whether you're rearranging from standard form, looking at a graph, or using two known points.