Step by step
The question
Question: find the equation of the line through (2, −1) and (4, 5).
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 (4, 5).
The first point is (2, −1).
Taking away a negative is the same as adding.
So it becomes (5 + 1)/(4 − 2).
5 plus 1 is 6.
4 minus 2 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 (2, −1): put x = 2 and y = −1 in.
3 times 2 is 6.
We want b by itself.
The 6 is in the way.
To remove 6, we take away 6.
We do it on both sides, so they stay equal.
6 and − 6 make zero.
Cross them out.
−1 minus 6 is −7.
So b = −7.
3 Write the equation
Put m = 3 and b = −7 into y = mx + b.
So the line is y = 3x − 7.
4 Check
Let's check: put x = 2 in.
We get −1, the y of (2, −1).
Now x = 4 in.
We get 5, the y of (4, 5).
Let's see it.
Both points are on the line y = 3x − 7.
And it crosses the y-axis at −7, just like b says.
Let's look back.
First: find the slope.
Then: find b.
Then: write the equation.
Last: check.