Step by step
The question
Question: find the line through (2, 3), perpendicular to x − 2y = 2.
1 Slope of the given line
The given line is not y = mx + b yet.
So we write it as y = mx + b first.
We want y by itself.
The x is in the way.
To remove x, we take away x.
We do it on both sides, so they stay equal.
x and − x make zero.
Cross them out.
Write what is left.
We write the x part first, like in y = mx + b.
Each part keeps its sign.
−2y means −2 times y.
We want just y.
To remove times −2, we divide by −2.
We do it on both sides, so they stay equal.
−2 divided by −2 is 1.
So just y is left.
The −2 divides the whole right side.
So it divides every part: −x and 2.
Each part keeps its sign on top.
Minus over minus is plus.
−x over −2 is the same as 1/2 times x.
Plus over minus is minus.
2 divided by −2 is −1.
So y = (1/2)x − 1.
m is the number in front of x: 1/2.
It is the slope.
2 Slope of the new line
Flip it: 2/1.
Then change the sign: −2.
So the new slope is −2.
Careful!
Not only the sign!
A line with slope −1/2 does not make a square corner.
Flip 1/2 too.
3 Find b
In y = mx + b, put m = −2.
Then use the point (2, 3): put x = 2 and y = 3 in.
Minus times plus is minus.
−2 times 2 is −4.
We want b by itself.
The − 4 is in the way.
To remove − 4, we add 4.
We do it on both sides, so they stay equal.
−4 and + 4 make zero.
Cross them out.
3 plus 4 is 7.
So b = 7.
4 Write the equation
Put m = −2 and b = 7 into y = mx + b.
So the new line is y = −2x + 7.
5 Check
Let's check: put x = 2 into y = −2x + 7.
We get 3, the y of (2, 3).
Another way to check: multiply them.
1/2 times −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 = −2x + 7 in red.
It goes through (2, 3).
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.