Step by step
The question
Question: find the equation of the line through (−2, 3) 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 (−2, 3).
Taking away a negative is the same as adding.
So it becomes (1 − 3)/(2 + 2).
1 minus 3 is −2.
2 plus 2 is 4.
We can make 2/4 simpler: divide top and bottom by 2.
The minus stays.
So m = −1/2.
2 Find b
In y = mx + b, put m = −1/2.
Then use the point (−2, 3): put x = −2 and y = 3 in.
Minus times minus is plus.
−1/2 times −2 is 1.
We want b by itself.
The 1 is in the way.
To remove 1, we take away 1.
We do it on both sides, so they stay equal.
1 and − 1 make zero.
Cross them out.
3 minus 1 is 2.
So b = 2.
3 Write the equation
Put m = −1/2 and b = 2 into y = mx + b.
So the line is y = −(1/2)x + 2.
4 Check
Let's check: put x = −2 in.
We get 3, the y of (−2, 3).
Now x = 2 in.
We get 1, the y of (2, 1).
Let's see it.
Both points are on the line y = −(1/2)x + 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.