Step by step
The question
Question: find the equation of the line through (3, −2) and (−3, 6).
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 (3, −2).
The first point is (−3, 6).
Taking away a negative is the same as adding.
So it becomes (−2 − 6)/(3 + 3).
−2 minus 6 is −8.
3 plus 3 is 6.
We can make 8/6 simpler: divide top and bottom by 2.
The minus stays.
So m = −4/3.
2 Find b
In y = mx + b, put m = −4/3.
Then use the point (−3, 6): put x = −3 and y = 6 in.
Minus times minus is plus.
−4/3 times −3 is 4.
We want b by itself.
The 4 is in the way.
To remove 4, we take away 4.
We do it on both sides, so they stay equal.
4 and − 4 make zero.
Cross them out.
6 minus 4 is 2.
So b = 2.
3 Write the equation
Put m = −4/3 and b = 2 into y = mx + b.
So the line is y = −(4/3)x + 2.
4 Check
Let's check: put x = −3 in.
We get 6, the y of (−3, 6).
Now x = 3 in.
We get −2, the y of (3, −2).
Let's see it.
Both points are on the line y = −(4/3)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.