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

Step by step

The question

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

y = 3x − 1

new line: through (3, 2)

perpendicular to this one

1 Slope of the given line

m is the number in front of x: 3.

It is the slope.

b is −1.

m = 3

b = −1

2 Slope of the new line

3 is 3/1.

Flip it: 1/3.

Then change the sign: −1/3.

3 = 31 → flip: 13

→ change the sign: −13

So the new slope is −1/3.

m = −13

Careful!

Not only the sign!

A line with slope −3 does not make a square corner.

Flip 3 too.

Careful: not m = −3 ✗

3 Find b

In y = mx + b, put m = −1/3.

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

2 = −13 × 3 + b

Minus times plus is minus.

−1/3 times 3 is −1.

2 = −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.

2 + 1 = −1 + b ~~+ 1~~

−1 and + 1 make zero.

Cross them out.

2 plus 1 is 3.

So b = 3.

3 = b

4 Write the equation

Put m = −1/3 and b = 3 into y = mx + b.

So the new line is y = −(1/3)x + 3.

y = −13x + 3

5 Check

Let's check: put x = 3 into y = −(1/3)x + 3.

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

y = −13(3) + 3 = −1 + 3 = 2

Another way to check: multiply them.

3 times −1/3 is −1.

Perpendicular slopes always multiply to −1.

3 × (−13) = −1

Let's see it on the grid.

The given line in blue.

Our new line y = −(1/3)x + 3 in red.

It goes through (3, 2).

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