Question

solve the following system by substitution:
3x + y - 2z = 22
x + 5y + z = 4
x = -3z

I've tried a few ways to solve, but I can't get it. Can anyone show me a way to solve?

Answer 1

Let us use the third equation first.
We have : \(\color{blue}{x = -3z}\) according to the third equation. Right?

Answer 2

Now, we will plug-in the value of x from third equation, in any of the two equations. Let us pick the second one as it will be simpler for us to go through it.

We have : \(x + 5y +z = 4\) , according to the second equation.

\(\color{blue}{-3z} + 5y + z = 4\)

Note that I plugged in the value of x from third equation into the second equation/

Can you simplify the above equation a bit now?

Answer 3

Okay, I solved for z but I just got \[y=\frac{ 2(z+2) }{ 5 }\]

Answer 4

Okay, so, you solved for y. Cool!

So, put this value of y : \( \color{red}{y = \cfrac{2(z+2)}{5} }\) and that of x : \(\color{blue}{x = -3z}\) in equation (1).

See what you get.

Answer 5

oops I meant I solved for y not z yes haha

okay, so I'm solving this for z:
3(-3z) + 2(z+2)/5 - 2z = 22
and I got z = -2. Am I on the right track?

Answer 6

Excellent! You're on the right track.
Now plug-in this value of z in : \(\color{blue}{x = -3z }\) and thus, you will get the value of x.

Answer 7

So x=6!

Answer 8

And now I can plug my x and z values back into the first equation to solve for my y value?

Answer 9

okay, from 3(6)+y-2(-2)=22 I got y=0

Answer 10

Just plug-in the value of z in : \(\color{red}{y = \cfrac{2(z+2)}{5}}\)

Yeah, that's right.
Great work.

Answer 11

Awesome! Thank you so much for everything, you made that seem way easier than I thought it was going to be!!

Answer 12

:) You're welcome @boldlygo ! Though, I just gave you a hint and you did the rest. So, applauds for you. Keep it up and welcome to OpenStudy!