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# Slope formula

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:
$\text{Slope}=\frac{\text{Change in y}}{\text{Change in x}}$
Let's draw a line through two general points $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$.
An expression for $\text{change in x}$ is ${x}_{2}-{x}_{1}$:
Similarly, an expression for $\text{change in y}$ is ${y}_{2}-{y}_{1}$:
Now we can write a general formula for slope:
$\text{Slope}=\frac{\text{Change in y}}{\text{Change in x}}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$
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 $\left(2,1\right)$ and $\left(4,7\right)$.
Step 1: Identify the values of ${x}_{1}$, ${x}_{2}$, ${y}_{1}$, and ${y}_{2}$.
${x}_{1}=2$
${y}_{1}=1$
${x}_{2}=4$
Step 2: Plug in these values to the slope formula to find the slope.
$\text{Slope}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}=\frac{7-1}{4-2}=\frac{6}{2}=3$
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.

## Using the slope formula walkthrough

Let's use the slope formula to find the slope of the line that goes through the points $\left(6,-3\right)$ and $\left(1,7\right)$.
Step 1: Identify the values of ${x}_{1}$, ${x}_{2}$, ${y}_{1}$, and ${y}_{2}$.
${x}_{1}=$
${y}_{1}=$
${x}_{2}=$
${y}_{2}=$

Step 2: Plug in these values to the slope formula to find the slope.
$\text{Slope}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}=$

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

## Let's practice!

1) Use the slope formula to find the slope of the line that goes through the points $\left(2,5\right)$ and $\left(6,8\right)$.

2) Use the slope formula to find the slope of the line that goes through the points $\left(2,-3\right)$ and $\left(-4,3\right)$.

3) Use the slope formula to find the slope of the line that goes through the points $\left(-5,-7\right)$ and $\left(-2,-1\right)$.

What happens in the slope formula when ${x}_{2}={x}_{1}$?
As a reminder, here is the slope formula:
$\text{Slope}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$
Feel free to discuss in the comments below!

## Want to join the conversation?

• Why is the slope formula y/x? Why not y-x or y+x?

Jack.
• Slope is something that is also referred to as the rate of change. For example, if you had a savings account that you deposited no money into initially but you deposit 20\$ weekly, your rate of change, or slope for this problem would be 20. This is because your x-value in this situation would be the number of weeks passed since you have created your bank account, and the y-value is how much money you have deposited into your account, fully. Since you are looking at the rate of change between the weeks, you divide the change in y per week, 20, by 1 for the number of weeks. I hope this somewhat answers your question.
I think that when x_2 = x_1 then the slope will become undefined because x_2 - x_1 equals zero. Therefore when you divide y_2 - y_1 it won't be possible.

Example

(5,10) (5,15)

x_1 = 5
x_2 = 5
y_1 =10
y_2 =15

5 - 5= 0

15-10=5

5/0= Undefined
• Yes, you are correct. The slope of any line through two different points with the same x-coordinate (that is, a vertical line) is always undefined, for the reason you stated.
• I think that when X2 = X1, the slope is undefined
• Yes! That is correct.
• why do i have a feeling that im going to die after i make it through slopes
• Because we all will die
• If I Get The Right Answer Then Why Do I Have To Simplify?
• Simplifying just makes it easier to read/understand. It makes it more "simple." Although both are equal, it is just easier to work with if it's simplified afterwards.
• Using the slope​ formula, find the slope of the line through the points (0,0) and(3,6) . Use pencil and paper. Explain how you can use mental math to find the slope of the line. The slope of the line is enter your response here. ​(Type an integer or a simplified​ fraction.)
• 0s make it easy because you end up with a proportional relationship where y/x = 6/3 or when you reduce and multiply by x, you get y=2x. Using the slope formula, m= (6-0)/(3-0) which is just m=6/3=2.
When x1 = x2, it means that x1-x2=0. So therefore the formula will simplify to y/0. And as x/0 is undefined, the slope should also be undefined.