Step by step
The question
Question: are 2x + 3y = 6 and 3x − 2y = 4 parallel, perpendicular or neither.
The slopes tell us which.
Same slope: parallel.
Flip the slope and change its sign: perpendicular.
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 and − 2x make zero.
Cross them out.
Write what is left.
We write the x part first, like in y = mx + b.
Each part keeps its sign.
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.
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.
Minus over plus is minus.
−2x/3 is the same as −2/3 times x.
6 divided by 3 is 2.
So y = −(2/3)x + 2.
m is the number in front of x: −2/3.
It is the slope.
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 and − 3x make zero.
Cross them out.
Write what is left.
We write the x part first, like in y = mx + b.
Each part keeps its sign.
−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.
−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.
Minus over minus is plus.
−3x over −2 is the same as 3/2 times x.
Plus over minus is minus.
4 divided by −2 is −2.
So y = (3/2)x − 2.
m is the number in front of x: 3/2.
It is the slope.
3 Compare the slopes
−2/3 and 3/2 are not the same.
So the lines are not parallel.
Flip it: −3/2.
Then change the sign: 3/2.
That is the second slope, 3/2!
Another way to check: multiply them.
−2/3 times 3/2 is −1.
Perpendicular slopes always multiply to −1.
So the lines are perpendicular: they meet at a square corner.
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.