Question

How many solutions are there to the following system of equations?
2x + 6y = 4
5x + 15y = 10

A. 0
B. 1
C. infinitely many
D. 2

Answer 1

\(\large\color{blue}{ 2x + 6y = 4}\)
\(\large\color{blue}{ 5x + 15y = 10}\)

divide the first equation by 2, and the second equation by 5

Answer 2

Like this?

2x + 6y divided into 2?

Answer 3

Will the first one be 1x + 3y = 3 ?

Answer 4

\(\large\color{blue}{ 2x ~~+~~ 6y ~~= ~~4}\)

\(\large\color{blue}{ \Downarrow}\) \(\large\color{blue}{ \Downarrow}\) \(\large\color{blue}{ \Downarrow}\)
\(\large\color{blue}{ \div2}\) \(\large\color{blue}{ \div2}\) \(\large\color{blue}{ \div2}\)

\(\large\color{purple}{ x ~~+~~ 3y ~~= ~~2}\)

Answer 5

do the same division, but by 5, for the second equation.

Answer 6

Ah-ha! :) Okay, thanks! Let me try to do the second one.

Answer 7

yes, go ahead:)

Answer 8

@SolomonZelman The second equation is: 2.5x + 7.5y = 5

Answer 9

you are dividing by 5, not by 2

Answer 10

Right! I forgot. :( The answer is: 1x + 3y = 2

Answer 11

if the equations are the same, then there is infinitely many solutions

Answer 12

yes

Answer 13

@SolomonZelman What do we do now?

Answer 14

Oh, I had not noticed that they had the same equations!

Answer 15

\(\large\color{purple}{x+3y=2}\), this is what both of your equations came to be equal to, right?

Answer 16

Right.

Answer 17

Oh, you get it, i c....

Answer 18

\(\large\color{purple}{x+3y=2}\)
\(\large\color{purple}{^{^{\Huge-}}~~x+3y=2}\)
\(\large\color{purple}{^{\text{______________}}}\)
\(\large\color{blue}{0=0}\)

Answer 19

how many solutions are there?

Answer 20

oh
0=0 is true for any x value, correct?

Answer 21

There are no solutions.

Answer 22

I think.

Answer 23

your system, after your divisions that we performed, is:

\(\large\color{purple}{x+3y=2}\)
\(\large\color{purple}{x+3y=2}\)

\(\large\color{purple}{x=2-3y}\)
\(\large\color{purple}{x=2-3y}\)

\(\large\color{purple}{x=x}\)

Answer 24

x=x has infinitely many solutions.

Answer 25

When ever you have 2 equations that are multiples of each other (and here we do so too, as we have showed), that means infinity of solutions.

Answer 26

Oh, I think that I understand a little bit better now. :)

Answer 27

two lines on top of each other, they will intersect infinite times

Answer 28

And so our answer is C. infinitely many.

Answer 29

yup

Answer 30

Well, thank you for your help and time. I have three more questions, so if I need help can I tag you to the question(s)? @SolomonZelman