Step by step
The question
Question: find the line through (−1, 4), perpendicular to y = −x + 2.
1 Slope of the given line
m is the number in front of x: −1.
It is the slope.
b is 2.
2 Slope of the new line
−1 is −1/1.
Flip it: −1/1.
Then change the sign: 1.
So the new slope is 1.
Careful!
Not only the sign!
A line with slope 1 does not make a square corner.
Flip −1 too.
3 Find b
In y = mx + b, put m = 1.
Then use the point (−1, 4): put x = −1 and y = 4 in.
Minus times plus is minus.
1 times −1 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.
4 plus 1 is 5.
So b = 5.
4 Write the equation
Put m = 1 and b = 5 into y = mx + b.
So the new line is y = x + 5.
5 Check
Let's check: put x = −1 into y = x + 5.
We get 4, the y of (−1, 4).
Another way to check: multiply them.
−1 times 1 is −1.
Perpendicular slopes always multiply to −1.
Let's see it on the grid.
The given line in blue.
Our new line y = x + 5 in red.
It goes through (−1, 4).
They meet at a square corner.
Let's look back.
First: slope of the given line.
Then: slope of the new line.
Then: find b.
Then: write the equation.
Last: check.