CCSS Math: 8.F.B.4
Learn how to write the slope formula from scratch and how to apply it to find the slope of a line from two points.
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:
Slope=Change in yChange in x\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}
Let's draw a line through two general points (x1,y1)(\greenD{x_1}, \goldD{y_1}) and (x2,y2)(\greenD{x_2}, \goldD{y_2}).
An expression for change in x\greenD{\text{change in x}} is x2x1\greenD{ x_2 -x_1}:
Most people have to stop and think about why this expression works. For example, think about if x1=3x_1 = 3 and x2=7x_2 = 7. Here is how we would find the change in xx:
=x2x1=73=4\begin{aligned} &\phantom{=}\greenD{x_2 - x_1} \\\\ &= 7 - 3 \\\\ &= 4 \end{aligned}
This makes sense because the distance from 33 to 77 is 44.
Similarly, an expression for change in y\goldD{\text{change in y}} is y2y1\goldD{y_2 - y_1}:
Now we can write a general formula for slope:
Slope=Change in yChange in x=y2y1x2x1\text{Slope} = \dfrac{\goldD{\text{Change in y}}}{{\greenD{\text{Change in x}}}}= \dfrac{\goldD{y_2 - y_1}}{\greenD{x_2 - x_1}}
That's it! We did it!

Using the slope formula

Let's use the slope formula to find the slope of the line that goes through the points (2,1)(2,1) and (4,7)(4, 7).
Step 1: Identify the values of x1x_1, x2x_2, y1y_1, and y2y_2.
x1=2x_1 = 2
y1=1y_1 = 1
x2=4x_2 = 4
y2=7        y_2 = 7 ~~~~~~~~
It's easier to see if we line up the points we have
(2,1)(2,1) and (4,7)(4,7)
vertically with the general points
(x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2).
We can also think of it like this:
x1x_\redD1 is the xx-coordinate of the 1st\redD1^\text{st} point.
y1y_\redD1 is the yy-coordinate of the 1st\redD1^\text{st} point.
x2x_\redD2 is the xx-coordinate of the 2nd\redD2^\text{nd} point.
y2y_\redD2 is the yy-coordinate of the 2nd\redD2^\text{nd} point.
Step 2: Plug in these values to the slope formula to find the slope.
Slope=y2y1x2x1=7142=62=3\text{Slope} = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{7 - 1}{4 - 2} = \dfrac62 = 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.

Using the slope formula walkthrough

Let's use the slope formula to find the slope of the line that goes through the points (6,3)(6,-3) and (1,7)(1, 7).
Step 1: Identify the values of x1x_1, x2x_2, y1y_1, and y2y_2.
x1=x_1 =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
y1=y_1 =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
x2=x_2 =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
y2=y_2 =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

x1=6x_1 = 6
y1=3y_1 = -3
x2=1x_2 = 1
y2=7y_2 = 7
Step 2: Plug in these values to the slope formula to find the slope.
Slope=y2y1x2x1=\text{Slope} = \dfrac{y_2 - y_1}{x_2 - x_1} =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Slope=y2y1x2x1=7(3)16=105=2\text{Slope} = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{7 - (-3)}{1 - 6} = \dfrac{10}{-5} = -2
Step 3: Gut check. Make sure this slope makes sense by thinking about the points on the coordinate plane.
Does this slope make sense?
Choose 1 answer:
Choose 1 answer:

Let's practice!

1) Use the slope formula to find the slope of the line that goes through the points (2,5)(2, 5) and (6,8)(6, 8).
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Slope=y2y1x2x1=8562=34\text{Slope} = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{8 - 5}{6 - 2} = \dfrac{3}{4}
2) Use the slope formula to find the slope of the line that goes through the points (2,3)(2, -3) and (4,3)(-4, 3).
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Slope=y2y1x2x1=3(3)42=66=1\text{Slope} = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{3 - (-3)}{-4 - 2} = \dfrac{6}{-6} = -1
3) Use the slope formula to find the slope of the line that goes through the points (5,7)(-5, -7) and (2,1)(-2, -1).
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Slope=y2y1x2x1=1(7)2(5)=63=2\text{Slope} = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{-1 - (-7)}{-2 - (-5)} = \dfrac{6}{3} = 2

Something to think about

What happens in the slope formula when x2=x1x_2 = x_1?
As a reminder, here is the slope formula:
Slope=y2y1x2x1\text{Slope}= \dfrac{y_2 - y_1}{x_2 - x_1}
Feel free to discuss in the comments below!
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