Question

Which of the following is a solution to the system of linear equations below?

-2x+3y=3
x-6y=12

A.
(–6, –3)
B.
(–3, –6)
C.
(3, 6)
D.
(6, 3)

Answer 1

the solution will be the pair of numbers which makes all of the equations in the system true.

for example, D is NOT correct:
\[-2x+3y=3\]\[x-6y=12\]substitute \(x=6,y=3\)
\[-2(6)+3(3) = 3\rightarrow -12+9=3\]\[6-6(3)=12\rightarrow 6-18=12\]Neither of those is correct.

Note that you can have a solution which satisfies one equation but not the other(s) — that is also not a solution to the system.

Answer 2

If you want to solve this system of equations, I would suggest multiplying the 2nd equation by 2, then adding it to the first equation. You should then have an equation only in terms of one of the variables, which you can solve.

Answer 3

Well i know A goes for the first one but not the other. unless i did my math wrong

Answer 4

I mean C

Answer 5

Why don't we just solve it?

\[-2x+3y=3\]\[x-6y=12\]Multiply the second equation by 2:
\[-2x+3y=3\]\[2x-12y=24\]Now add them together
\[-2x+2x+3y-12y = 3+24\]\[0x-9y=27\]\[-9y=27\]You can solve that for \(y\), I'm sure!

Answer 6

Now you take that value of \(y\) and plug it into either of the original equations and solve for the other variable. Bam, you're done.

Answer 7

You just need to look at the equations and see how much to multiply one equation by to make one of the terms have an equal but opposite coefficient to the corresponding term in the other equation.

Answer 8

So is the answer A?

Answer 9

That process is called elimination. It's also easy to solve this one by substitution:

\[-2x+3y=3\]\[x-6y=12\]Solve one of the equations for one of the variables in terms of the other. Here the obvious thing is to solve the second one for \(x\) because it's almost there already!
\[x-6y=12\]\[x=12+6y\]Now we go back to the first equation and substitute \((12+6y)\) wherever we see \(x\)
\[-2(12+6y)+3y=3\]Solving for \(y\)
\[-24-12y+3y=3\]\[-24-9y=3\]\[-9y=27\]\[y=-3\]
Again, take that value and find the other one:
\[x=12+6y\]\[x=12+6(-3)\]\[x=12-18\]\[x=-6\]

Answer 10

Yes, A is the correct answer because we found x=-6, y=-3

Answer 11

Okay thank you that helped a lot :D

Answer 12

but as I said earlier, it is important to check your answer in all of the equations to make sure it works everywhere. let's do so:

\[-2x+3y=3\]\[-2(-6)+3(-3) = 3\]\[12-9=3\checkmark\]
\[x-6y=12\]\[-6-6(-3)=12\]\[-6+18=12\checkmark\]Okay, they both pass the test, our solution is good.

Answer 13

Yay! :)

Answer 14

you could have done this problem just by checking each answer and eliminating all the wrong ones. as it turns out, you would have gotten lucky with the first answer :-) but better to know how to do these!

Answer 15

with practice, you'll be able to do these in your head faster than I was able to type them.