Slope formula (equation for slope) | Algebra (article) | Khan Academy (2024)

It's kind of annoying to have to draw a graph every time we want to find the slope of a line, isn't it?

We can avoid this by writing a general formula for slope. Before we start, let's remember how slope is defined:

$\text{Slope}={\displaystyle \frac{\text{Change in y}}{\text{Change in x}}}$‍

Let's draw a line through two general points $(x_1,y_1)$‍ and $(x_2,y_2)$‍.

Similarly, an expression for $\text{change in y}$‍ is $y_2-y_1$‍:

Now we can write a general formula for slope:

$\text{Slope}={\displaystyle \frac{\text{Change in y}}{\text{Change in x}}}={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$‍

That's it! We did it!

Let's use the slope formula to find the slope of the line that goes through the points $(2,1)$‍ and $(4,7)$‍.

Step 1: Identify the values of $x_1$‍, $x_2$‍, $y_1$‍, and $y_2$‍.

$x_1=2$‍

$y_1=1$‍

$x_2=4$‍

$y_2=7$‍

It's easier to see if we line up the points we have

$(2,1)$‍ and $(4,7)$‍

vertically with the general points

$(x_1,y_1)$‍ and $(x_2,y_2)$‍.

We can also think of it like this:

$x_1$‍ is the $x$‍-coordinate of the $1^{\text{st}}$‍ point.

$y_1$‍ is the $y$‍-coordinate of the $1^{\text{st}}$‍ point.

$x_2$‍ is the $x$‍-coordinate of the $2^{\text{nd}}$‍ point.

$y_2$‍ is the $y$‍-coordinate of the $2^{\text{nd}}$‍ point.

Step 2: Plug in these values to the slope formula to find the slope.

$\text{Slope}={\displaystyle \frac{y_2-y_1}{x_2-x_1}}={\displaystyle \frac{7-1}{4-2}}={\displaystyle \frac{6}{2}}=3$‍

Step 3: Gut check. Make sure this slope makes sense by thinking about the points on the coordinate plane.

Yup! This slope seems to make sense since the slope is positive, and the line is increasing.

Let's use the slope formula to find the slope of the line that goes through the points $(6,-3)$‍ and $(1,7)$‍.

Step 1: Identify the values of $x_1$‍, $x_2$‍, $y_1$‍, and $y_2$‍.

Step 2: Plug in these values to the slope formula to find the slope.

Step 3: Gut check. Make sure this slope makes sense by thinking about the points on the coordinate plane.

What happens in the slope formula when $x_2=x_1$‍?

As a reminder, here is the slope formula:

$\text{Slope}={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$‍

Feel free to discuss in the comments below!