Are 2x + y = 4 and y = -2x + 1 parallel, perpendicular or neither Step by Step

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The question

Question: are 2x + y = 4 and y = −2x + 1 parallel, perpendicular or neither.

① 2x + y = 4 and ② y = −2x + 1

The slopes tell us which.

Same slope: parallel.

Flip the slope and change its sign: perpendicular.

parallel: same slope

perpendicular: flip and change the sign

1 Slope of ①

① is not y = mx + b yet.

So we write it as y = mx + b first.

We want y by itself.

The 2x is in the way.

To remove 2x, we take away 2x.

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

2x + y ~~− 2x~~ = 4 − 2x

2x and − 2x make zero.

Cross them out.

Write what is left.

y = 4 − 2x

We write the x part first, like in y = mx + b.

Each part keeps its sign.

y is by itself.

y = −2x + 4

m is the number in front of x: −2.

It is the slope.

m₁ = −2

b = 4

2 Slope of ②

m is the number in front of x: −2.

It is the slope.

b is 1.

m₂ = −2

b = 1

3 Compare the slopes

Both slopes are −2.

The same steepness.

−2 = −2

But they start at different places: 4 and 1.

b = 4 and b = 1

So the lines are parallel: they never meet.

① and ② are parallel

4 Check

Let's see it on the grid.

Draw both lines.

They never meet.

Let's look back.

First: slope of ①.

Then: slope of ②.

Then: compare the slopes.

Last: check.

Answer