Step by step
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