Question

I got the answer by guessing but need to learn the steps to get the answer. Can some one walk me through this please?

A software designer is mapping the streets for a new racing game. All the streets are depicted as either perpendicular or parallel lines. The equation of the line passing through A and B is -7x+3y=-21.5. What is the equation of PQ?

The answer is -1.5x-3.5y=-31.5

Answer 1

Welcome to the world of "LINES" binks :3

How can I help you? ^_^

Answer 2

I know the answer because I guessed but really i dont understand this

Answer 3

Yes, so basically, what we have is a simple "find the equation of a line" problem, right? ^_^

Answer 4

yes

Answer 5

Okay, we either get a line from knowing two points on the line, OR one point and the slope of the line.

Catch me so far?

Answer 6

yes

Answer 7

Are you sure?
Okay, do you know what/how to get the slope of a line? ^_^

Answer 8

yes

Answer 9

That's great :)
Then what's the slope of the line through AB?

Answer 10

well if you sign up for mathway.com then you can get the solution and how they did the problem.

Answer 11

1/3

Answer 12

Such isn't the case :)
Here, let me give you a subtle hint.
When you have an equation
\[\Large \color{red}Ax+\color{green}By= \color{blue}C\]
Then the slope of the line is given by:
\[\Large m = -\frac{\color{red}A}{\color{green}B}\]

Answer 13

m=2/6 m=1/3

Answer 14

how do you get 2/6 ?

Answer 15

B (9,6)
C (3,4)

Answer 16

Oh, I see, a different formula, then?
Okay, fine, but you used points B and C, when I asked for the line through A and B :)

And that method's not going to help you... refer to the formula I posted....

Answer 17

I dont know what numbers Im suppose to imput

Answer 18

input :P

Answer 19

That's because you're not given the precise coordinates of points A and B.
What you ARE given is the EQUATION of the line through A and B, and you know what, that's good enough if you just want to find the slope ^_^

PLEASE refer to the formula I posted :D

Answer 20

m=7/3

Answer 21

That's good.
Now, we don't know squat about the slope of PQ yet, but we do know that PQ is perpendicular to AB.

What does that say about the slope of PQ?

Answer 22

not sure sorry

Answer 23

m=-3/7

Answer 24

-3x/7

Answer 25

Correct slope.
Nicely done.
So, you know the slope of line PQ, do we know of any POINT that lies on PQ?

Answer 26

they intersect at 10.5

Answer 27

No... don't be too hasty... we just need a POINT on PQ.
I believe one of them is marked out in your attached picture...

Answer 28

(7,6)

Answer 29

Okay, good :)
When you have a point (a,b) and a slope m, then the equation of the line is always given by...
\[\Large y = \color{blue}m(x-\color{red}a)+\color{green}b\]

Answer 30

so my point is (7,6) and my slope is m--3x/7 am I right so far

Answer 31

slope should be a pure number, no x.

Answer 32

typo oh ok so m=3x/7

Answer 33

no, you still have an x... m should not have x in it.

Answer 34

y=3/7*x+b

Answer 35

Yes, but what's b?
That's the second coordinate of your known point.

Answer 36

no...

Answer 37

b= 6

Answer 38

Good. Now plug in :)

Answer 39

But make sure to plug it in correctly, ok?
\[\Large y = m(x-a)+b\]

Answer 40

y=3/7*7+6
now solve for y

Answer 41

Wait what?
Whoa there... just replace a and b first...

Answer 42

y=3/7(x-7)+6

Answer 43

Okay, now simplify...

Answer 44

y=3(x+7)/7

Answer 45

Try again.... use the draw option, if you want.

Answer 46

y=3x+21/7

Answer 47

Oh okay then, multiply everything by 7, to get rid of that denominator.

Answer 48

y=3x+21

Answer 49

Where's the 7?
Surely y should have a 7

Answer 50

im sorry im getting more confused

Answer 51

7y = 3x + 21

Answer 52

Because we should multiply BOTH sides of the equation by 7.

Answer 53

I keep getting the same thing
y=3x/7 + 21/7

Answer 54

its ok thanks for trying im an idiot

Answer 55

no you're not -_-
But it's just that we're trying to learn in an hour what normally takes days, so don't get frustrated, and try again :)

Answer 56

7(y)=3x/7 + 21/7 and then I divide

Answer 57

No, the 7's should disappear from the right-side.

Answer 58

y=3x/7+3

Answer 59

@texaschic101 can you help me with the steps please?

Answer 60

notice that line AB is perpendicular to line PQ and the equation for line AB is -7x + 3y =-21.5. So we need to find the slope of this line by putting it in y = mx + b form, and m is the slope.
-7x + 3y = - 21.5 -- add 7x to both sides
3y = 7x - 21.5 -- divide by 3 on both sides
y = 7/3x - 21.5/3
the slope in this equation is 7/3.
A perpendicular line will have a negative reciprocal slope. All that means is " flip " the slope and change the sign. So the negative reciprocal slope of 7/3 is -3/7 (see how I flipped the slope and changed the sign).

Now I am going to use y = mx + b
slope(m) = -3/7
(7,6) x = 7 and y = 6
now we sub and solve for b
6 = -3/7(7) + b
6 = -3 + b
6 + 3 = b
9 = b
equation in slope intercept is : y = -3/7x + 9
but it appears you need it in standard form Ax + By = C
y = -3/7x + 9 --- multiply the entire equation by 7
7y = -3x + 63 -- add 3x to both sides
3x + 7y = 63

I am getting something totally different...let me try something else
y - y1 = m(x - x1)
slope(m) = -3/7
(7,6) x1 = 7 and y1 = 6
y - 6 = -3/7(x - 7)
y - 6 = -3/7x + 3
y = -3/7x + 3 + 6
y = -3/7x + 9
nope...I am gonna get the same answer and it is not your answer choice.
I am sorry but I am not seeing my mistake.

oh....quick question.....is the equation supposed to be -7x + 3y = -21.5 ??

Answer 61

yes -7x + 3y = -21.5 just doubled checked. Thats why I so confused. :(

Answer 62

Thanks for trying!!!

Answer 63

@whpalmer4

Answer 64

So, you know that the line passing through A and B is \(-7x+3y=-21.5\), and it is perpendicular to the line passing through P(7,6) and Q. First, let's find the slope of the line through A and B:
\[m = \frac{y_2-y_1}{x_2-x_1} = \frac{7}3\](gotten by visual inspection of the graph: from A, we go right 3 and up 7 to get to B, with each square representing 0.5 units )

Now, our line from P to Q is perpendicular to this line, so it has a slope which is the negative reciprocal \((-1/m)\):

\[m_{PQ} = -\frac{1}{\frac{7}3} = -\frac{3}{7}\]

With our slope and a known point, we use the point-slope equation to find the equation of the line through PQ:
\[y-y_0 = m(x-x_0)\]\[y-y_0 = -\frac{3}{7}(x-x_0)\]
Our known point is P(7,6), so
\[y-6 = -\frac{3}7(x-7)\]or\[7y-42=-3x+21\]\[7y+3x=63\]\[\frac{7}{2}y+\frac{3}{2}x=\frac{63}{2}\]\[3.5y+1.5x=31.5\]

Answer 65

So then you divide by 2

Answer 66

@texaschic101 You were on the right track! Last thing you had was
\[y = -\frac{3}{7}x + 9\]Multiply through by 7\[7y=-3x+63\]\[7y+3x=63\]Divide by 2\[3.5y+1.5x=31.5\]

Answer 67

oh wow thanks you guys you are awesome. need to write these steps down. Phew!!!!

Answer 68

yeah I see now I wasn't multiply right