Step by step
The question
Question: find the equation of the line through (−1, 5) and (1, −1).
Every line that is not straight up and down is y = mx + b.
m is the slope.
b is where the line crosses the y-axis: where we start.
So we need m and b.
1 Find the slope
We call the slope m.
m = rise over run.
The top is the change in y: second y minus first y.
The bottom is the change in x: second x minus first x.
Put the numbers in.
The second point is (1, −1).
The first point is (−1, 5).
Taking away a negative is the same as adding.
So it becomes (−1 − 5)/(1 + 1).
−1 minus 5 is −6.
1 plus 1 is 2.
−6 divided by 2 is −3.
So m = −3.
2 Find b
In y = mx + b, put m = −3.
Then use the point (−1, 5): put x = −1 and y = 5 in.
Minus times minus is plus.
−3 times −1 is 3.
We want b by itself.
The 3 is in the way.
To remove 3, we take away 3.
We do it on both sides, so they stay equal.
3 and − 3 make zero.
Cross them out.
5 minus 3 is 2.
So b = 2.
3 Write the equation
Put m = −3 and b = 2 into y = mx + b.
So the line is y = −3x + 2.
4 Check
Let's check: put x = −1 in.
We get 5, the y of (−1, 5).
Now x = 1 in.
We get −1, the y of (1, −1).
Let's see it.
Both points are on the line y = −3x + 2.
And it crosses the y-axis at 2, just like b says.
Let's look back.
First: find the slope.
Then: find b.
Then: write the equation.
Last: check.