Line through (3, 1) perpendicular to y = x + 4 Step by Step

Step by step

The question

Question: find the line through (3, 1), perpendicular to y = x + 4.

y = x + 4

new line: through (3, 1)

perpendicular to this one

1 Slope of the given line

m is the number in front of x: 1.

It is the slope.

b is 4.

m = 1

b = 4

2 Slope of the new line

1 is 1/1.

Flip it: 1/1.

Then change the sign: −1.

1 = 11 → flip: 11

→ change the sign: −1

So the new slope is −1.

m = −1

Careful!

Not only the sign!

A line with slope −1 does not make a square corner.

Flip 1 too.

Careful: not m = −1 ✗

3 Find b

In y = mx + b, put m = −1.

Then use the point (3, 1): put x = 3 and y = 1 in.

1 = −1 × 3 + b

Minus times plus is minus.

−1 times 3 is −3.

1 = −3 + b

We want b by itself.

The − 3 is in the way.

To remove − 3, we add 3.

We do it on both sides, so they stay equal.

1 + 3 = −3 + b ~~+ 3~~

−3 and + 3 make zero.

Cross them out.

1 plus 3 is 4.

So b = 4.

4 = b

4 Write the equation

Put m = −1 and b = 4 into y = mx + b.

So the new line is y = −x + 4.

y = −x + 4

5 Check

Let's check: put x = 3 into y = −x + 4.

We get 1, the y of (3, 1).

y = −3 + 4 = 1

Another way to check: multiply them.

1 times −1 is −1.

Perpendicular slopes always multiply to −1.

1 × (−1) = −1

Let's see it on the grid.

The given line in blue.

Our new line y = −x + 4 in red.

It goes through (3, 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.

Answer