The question
Question: are y = 2x + 1 and y = 2x − 3 parallel, perpendicular or neither.
① y = 2x + 1 and ② y = 2x − 3
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
m is the number in front of x: 2.
It is the slope.
b is 1.
m1 = 2
b = 1
2 Slope of ②
m is the number in front of x: 2.
It is the slope.
b is −3.
m2 = 2
b = −3
3 Compare the slopes
Both slopes are 2.
The same steepness.
2 = 2
But they start at different places: 1 and −3.
b = 1 and b = −3
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 ① and ② are parallel
▶ Watch it solved
The question
Question: are y = (2/3)x + 1 and y = −(3/2)x − 2 parallel, perpendicular or neither.
① y = 23x + 1 and ② y = −32x − 2
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
m is the number in front of x: 2/3.
It is the slope.
b is 1.
m1 = 23
b = 1
2 Slope of ②
m is the number in front of x: −3/2.
It is the slope.
b is −2.
m2 = −32
b = −2
3 Compare the slopes
2/3 and −3/2 are not the same.
So the lines are not parallel.
23 ≠ −32
Flip it: 3/2.
Then change the sign: −3/2.
23 → flip: 32
→ change the sign: −32
That is the second slope, −3/2!
−32 = −32
Another way to check: multiply them.
2/3 times −3/2 is −1.
Perpendicular slopes always multiply to −1.
23 × (−32) = −1
perpendicular slopes multiply to −1
So the lines are perpendicular: they meet at a square corner.
① and ② are perpendicular
4 Check
Let's see it on the grid.
Draw both lines.
They meet at a square corner.
Let's look back.
First: slope of ①.
Then: slope of ②.
Then: compare the slopes.
Last: check.
Answer ① and ② are perpendicular
▶ Watch it solved
The question
Question: are 2x + 3y = 6 and 3x − 2y = 4 parallel, perpendicular or neither.
① 2x + 3y = 6 and ② 3x − 2y = 4
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 + 3y − 2x = 6 − 2x
2x and − 2x make zero.
Cross them out.
Write what is left.
3y = 6 − 2x
We write the x part first, like in y = mx + b.
Each part keeps its sign.
3y = −2x + 6
3y means 3 times y.
We want just y.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3y3 = −2x + 63
3 divided by 3 is 1.
So just y is left.
The 3 divides the whole right side.
So it divides every part: −2x and 6.
Each part keeps its sign on top.
y = −2x3 + 63
Minus over plus is minus.
−2x/3 is the same as −2/3 times x.
−2x3 = −23x
6 divided by 3 is 2.
63 = 2
So y = −(2/3)x + 2.
y = −23x + 2
m is the number in front of x: −2/3.
It is the slope.
m1 = −23
b = 2
2 Slope of ②
② is not y = mx + b yet.
So we write it as y = mx + b first.
We want y by itself.
The 3x is in the way.
To remove 3x, we take away 3x.
We do it on both sides, so they stay equal.
3x − 2y − 3x = 4 − 3x
3x and − 3x make zero.
Cross them out.
Write what is left.
−2y = 4 − 3x
We write the x part first, like in y = mx + b.
Each part keeps its sign.
−2y = −3x + 4
−2y means −2 times y.
We want just y.
To remove times −2, we divide by −2.
We do it on both sides, so they stay equal.
−2y−2 = −3x + 4−2
−2 divided by −2 is 1.
So just y is left.
The −2 divides the whole right side.
So it divides every part: −3x and 4.
Each part keeps its sign on top.
y = −3x−2 + 4−2
Minus over minus is plus.
−3x over −2 is the same as 3/2 times x.
−3x−2 = 32x
Plus over minus is minus.
4 divided by −2 is −2.
4−2 = −2
So y = (3/2)x − 2.
y = 32x − 2
m is the number in front of x: 3/2.
It is the slope.
m2 = 32
b = −2
3 Compare the slopes
−2/3 and 3/2 are not the same.
So the lines are not parallel.
−23 ≠ 32
Flip it: −3/2.
Then change the sign: 3/2.
−23 → flip: −32
→ change the sign: 32
That is the second slope, 3/2!
32 = 32
Another way to check: multiply them.
−2/3 times 3/2 is −1.
Perpendicular slopes always multiply to −1.
−23 × 32 = −1
perpendicular slopes multiply to −1
So the lines are perpendicular: they meet at a square corner.
① and ② are perpendicular
4 Check
Let's see it on the grid.
Draw both lines.
They meet at a square corner.
Let's look back.
First: slope of ①.
Then: slope of ②.
Then: compare the slopes.
Last: check.
Answer ① and ② are perpendicular
▶ Watch it solved
