5. Determine if the lines represented by 2x + 3y = 15 and y = 3/2x – 6 are parallel, perpendicular, neither, or the same line.

Question

mathstudent55 · Answer

What do you know about the slopes of parallel lines?

ellerai · Answer

that they are positive?

mathstudent55 · Answer

No. Slopes of lines in general can be positive, negative, zero, or even undefined (when there is division by 0).
Parallel lines have the same slope.
The slopes of parallel lines are equal.
Ok?

ellerai · Answer

yes

mathstudent55 · Answer

Next. What about the slopes of perpendicular lines?

ellerai · Answer

they are the opposite of the slope

mathstudent55 · Answer

The slopes of perpendicular lines are negative reciprocals.
That means if two lines are perpendicular, their slopes have a product of -1.
That also means, if you see the slope of a line, then the slope of a perpendicular to that line is obtained by flipping the fraction and changing the sign of the slope of the first line.

ellerai · Answer

okay

mathstudent55 · Answer

Examples:

parallel lines: two lines have the slope 2/3

perpendicular lines: one line has slope 2/3, and the other line has slope -3/2

ellerai · Answer

yeah

mathstudent55 · Answer

Now let's look at your problem.
Look at the equation of the second line

$y = \dfrac{3}{2} x - 6$

This equation is very convenient for our problem because it is in the slope-intercept form.

When you write the equation of a line in the slope-intercept form it looks like this:

$y = mx + b$, 

and m is the slope.

You can read directly from your second equation that the slope of the second line is 3/2.

ellerai · Answer

yep :)

mathstudent55 · Answer

We already know what the slope of the second line is.
Now we need to find out what the slope f the first line is to see if it is the same as this one, making the lines parallel, or if it is the negative reciprocal of this one, making the lines perpendicular, or if it is neither.

mathstudent55 · Answer

Let's take the equation of the first line, and put it in the slope-intercept form,

$y =- mx + b$

ellerai · Answer

Okay how do we turn it into slope intercept form tho

mathstudent55 · Answer

We need to solve the equation of the first line for y.

ellerai · Answer

2x + 3y = 15 so would we isolate y?

ellerai · Answer

by subtracting 2x from both sides?

mathstudent55 · Answer

$2x + 3y = 15$

We want y alone on the left side.
2x is being added to the left side, so we get rid of the addition of 2x by subtracting 2x from both sides.

mathstudent55 · Answer

Exactly. You're faster than me.
Pretty soon, I'll be asking you for math help.

ellerai · Answer

Lol :P nah im bad at math

mathstudent55 · Answer

$2x + 3y = 15$

$2x -2x + 3y = -2x + 15$

$3y = -2x + 15$

So far, so good.
The next step is that we have 3y, but we want just y on the left side.
Since the 3 is multiplying y, we divide both sides by 3.

mathstudent55 · Answer

$3y = -2x + 15$

$\dfrac{3y}{3} = -\dfrac{2}{3}x + \dfrac{15}{3}$

Now we simplify the fractions.

ellerai · Answer

y = -2/3x + 5?

mathstudent55 · Answer

$y = -\dfrac{2}{3}x + 5$

Correct.

mathstudent55 · Answer

Now we have the slope of the second line.
What is it?

ellerai · Answer

-2/3

mathstudent55 · Answer

Correct.
Now compare slopes: $\dfrac{3}{2} $ and  $-\dfrac{2}{3} $.

They are clearly not the same. The lines are definitely not parallel.

Are they negative reciprocals of each other making the lines perpendicular?

ellerai · Answer

yes, because 3/2 as a negative reciprocal is -2/3

mathstudent55 · Answer

Correct.
When you flip one of the slopes and change its sign you get the other.
More formally in math, when you multiply them both together, you get -1.
Both of the above make the slopes negative reciprocals, and the lines are perpendicular.

mathstudent55 · Answer

Great job!

ellerai · Answer

yay! thank you so much! !

mathstudent55 · Answer

You're very welcome.