Step by step
The question
Question: find the equation of the line through (2, 1) 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, −1).
Taking away a negative is the same as adding.
So it becomes (1 + 1)/(2 + 2).
1 plus 1 is 2.
2 plus 2 is 4.
We can make 2/4 simpler: divide top and bottom by 2.
So m = 1/2.
2 Find b
In y = mx + b, put m = 1/2.
Then use the point (−2, −1): put x = −2 and y = −1 in.
Minus times plus is minus.
1/2 times −2 is −1.
We want b by itself.
The − 1 is in the way.
To remove − 1, we add 1.
We do it on both sides, so they stay equal.
−1 and + 1 make zero.
Cross them out.
−1 plus 1 is 0.
So b = 0.
3 Write the equation
Put m = 1/2 and b = 0 into y = mx + b.
So the line is y = (1/2)x.
4 Check
Let's check: put x = −2 in.
We get −1, the y of (−2, −1).
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.
And it crosses the y-axis at 0, just like b says.
Let's look back.
First: find the slope.
Then: find b.
Then: write the equation.
Last: check.