Question

The coordinates below represent two linear equations.

How many solutions does this system of equations have?


Line 1 Line 2
x y x y
-3 6 -6 7
3 4 9 2
A. 0

B. exactly 1

C. exactly 2

D. infinitely many

Answer 1

Have you considered calculating the respective slopes?

Answer 2

I have both of their slopes are \[\frac{ 1 }{ 3 }\]

Answer 3

Okay, that narrows it down to Zero (0) or Infinitely many. One (1) is no longer an option.

Now what?

Answer 4

BTW

c. Exactly 2 is just silly. Never pick that for linear equations.

Answer 5

You may also wish to figure out why you managed a positive slope when both are negative.

Answer 6

\[\frac{ y _{1}-y _{2} }{ x _{1}-x _{2} }\]

Answer 7

Indeed. Somehow, something went wrong in there.

Answer 8

Line 1
\[\frac{ 6-4 }{ -3-3 }\]

Answer 9

\(\dfrac{6-4}{-3-3} = \dfrac{2}{-6} = -\dfrac{1}{3}\)

Answer 10

Oh right I forgot to include the - but it was in my notebook

Answer 11

Okay, now that we have the right equal slopes, we'll need the y-intercepts, as well.

Are they the same line? Infinitely many solutions.
Are tehy different lines? Zero solutions.

Answer 12

I think the slope fr line 1 is 4 and the slope for line 2 is 2

Answer 13

Is that right? I'm not sure.

Answer 14

?? We already established that the slopes of both are -1/3.

What are these "4" and "2"?

Answer 15

Using the Point-Slope Form:

y-6 = (-1/3)(x+3)
y-7 = (-1/3)(x+6)

Same or different?

Answer 16

I meant the y-intercept

Answer 17

They are different

Answer 18

y-6 = (-1/3)(x+3)
y-7 = (-1/3)(x+6)

-3y+18 = x+3
-3y+21 = x+6

15 = x + 3y
15 = x + 3y

They look kind of the same to me. Why did you get a different result? You weren't being sloppy, were you?

Answer 19

Oh, I messed up

Answer 20

Thank you, you helped a ton!