Step by step
The question
Question: find the line through (4, 1), perpendicular to y = 2x − 5.
1 Slope of the given line
m is the number in front of x: 2.
It is the slope.
b is −5.
2 Slope of the new line
2 is 2/1.
Flip it: 1/2.
Then change the sign: −1/2.
So the new slope is −1/2.
Careful!
Not only the sign!
A line with slope −2 does not make a square corner.
Flip 2 too.
The right way: m = −1/2.
3 Find b
In y = mx + b, put m = −1/2.
The point (4, 1) is on the line, so its x and y fit.
We put x = 4 and y = 1 in.
Minus times plus is minus.
−1/2 times 4 is −2.
Is b alone now?
Not yet: − 2 is still with b.
How do we remove it?
We want b by itself.
The − 2 is in the way.
To remove − 2, we add 2.
We do it on both sides, so they stay equal.
−2 and + 2 cancel each other.
They make zero.
1 plus 2 is 3.
So b = 3.
So after step 3: we added 2, and now we have 3 = b.
4 Write the equation
Put m = −1/2 and b = 3 into y = mx + b.
So the new line is y = −(1/2)x + 3.
5 Check
Let's check: put x = 4 into y = −(1/2)x + 3.
We get 1, the y of (4, 1).
Another way to check: multiply them.
2 times −1/2 is −1.
Perpendicular slopes always multiply to −1.
Let's see it on the grid.
The given line in blue.
Our new line y = −(1/2)x + 3 in red.
It goes through (4, 1).
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.