Line through (2, 3) perpendicular to y = 2x

Step by step

The question

Question: find the line through (2, 3), perpendicular to y = 2x − 1.

y = 2x − 1

new line: through (2, 3)

perpendicular to this one

1 Slope of the given line

m is the number in front of x: 2.

It is the slope.

b is −1.

m = 2

b = −1

2 Slope of the new line

2 is 2/1.

Flip it: 1/2.

Then change the sign: −1/2.

2 = 21 → flip: 12

→ change the sign: −12

So the new slope is −1/2.

m = −12

Careful!

Not only the sign!

A line with slope −2 does not make a square corner.

Flip 2 too.

Careful: not m = −2 ✗

3 Find b

In y = mx + b, put m = −1/2.

Then use the point (2, 3): put x = 2 and y = 3 in.

3 = −12 × 2 + b

Minus times plus is minus.

−1/2 times 2 is −1.

3 = −1 + b

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.

3 + 1 = −1 + b ~~+ 1~~

−1 and + 1 make zero.

Cross them out.

3 plus 1 is 4.

So b = 4.

4 = b

4 Write the equation

Put m = −1/2 and b = 4 into y = mx + b.

So the new line is y = −(1/2)x + 4.

y = −12x + 4

5 Check

Let's check: put x = 2 into y = −(1/2)x + 4.

We get 3, the y of (2, 3).

y = −12(2) + 4 = −1 + 4 = 3

Another way to check: multiply them.

2 times −1/2 is −1.

Perpendicular slopes always multiply to −1.

2 × (−12) = −1

Let's see it on the grid.

The given line in blue.

Our new line y = −(1/2)x + 4 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.

Answer