Line through (3, 2) perpendicular to y = -3x + 1 Step by Step

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

1/3 times 3 is 1.

2 = 1 + b

We want b by itself.

The 1 is in the way.

To remove 1, we take away 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 minus 1 is 1.

So b = 1.

1 = b

4 Write the equation

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

So the new line is y = (1/3)x + 1.

y = 13x + 1

5 Check

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

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

y = 13(3) + 1 = 1 + 1 = 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 + 1 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