Step by step
The question
Question: find the equation of the line through (−1, −5) and (2, 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 (2, 1).
The first point is (−1, −5).
Taking away a negative is the same as adding.
So it becomes (1 + 5)/(2 + 1).
1 plus 5 is 6.
2 plus 1 is 3.
6 divided by 3 is 2.
So m = 2.
2 Find b
In y = mx + b, put m = 2.
Then use the point (−1, −5): put x = −1 and y = −5 in.
Minus times plus is minus.
2 times −1 is −2.
We want b by itself.
The − 2 is in the way.
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
−2 and + 2 make zero.
Cross them out.
−5 plus 2 is −3.
So b = −3.
3 Write the equation
Put m = 2 and b = −3 into y = mx + b.
So the line is y = 2x − 3.
4 Check
Let's check: put x = −1 in.
We get −5, the y of (−1, −5).
Now x = 2 in.
We get 1, the y of (2, 1).
Let's see it.
Both points are on the line y = 2x − 3.
And it crosses the y-axis at −3, just like b says.
Let's look back.
First: find the slope.
Then: find b.
Then: write the equation.
Last: check.