Question

7x-y-5z=5
6x+y-z=7
3x+y-7x=7
need help solving, please explain steps

Answer 1

In the last equation, is that \(3x+y-7z=7\)?

Answer 2

yes

Answer 3

First, in the process of solving it, notice that equation 1 has a \(-y\) term, while equation 2 has \(+y\) term. Thus, if you add the equations together you get a \(0y =0\) term where the \(y\) used to be.

Answer 4

\[\begin{matrix}7x-y-5z=5\\6x+y-z=7\\ \text{______________}\\13x-6z=12\end{matrix}\]

Answer 5

Let's call this equation 4.

Answer 6

My apologies. We don't want to get rid of x, we want to get rid of y here. Fortunately, both equation 2 and 3 have a \(+y\) term so we just subtract the two equations.\[\begin{matrix}6x+y-z=7\\ - \quad 3x+y-7z=-7\\ \text{______________________}\\3x+6z=14\end{matrix}\]

Answer 7

Let's call this equation 5.

Answer 8

Now we see that equation 4 has a \(-6z\) term, and equation 5 has a \(+6z\) term. Thus we can add them again.\[\begin{matrix}\:\:\quad13x-6z=12\\ - \quad 3x+6z=14\\ \text{______________________}\\16x=26\end{matrix}\]From here we can easily solve for \(x=26/16=13/8\)

Answer 9

Plug this value for x back into equation 5. \[3(13/8)+6z=14\]So we can find that \(z=73/48\)

Answer 10

If we plug both of these values into equation 2 (any of the first 3 should work) we get \[6(13/8) +y -(73/48)=7\]We finally get that \(y=175/48\)

Answer 11

I can solve for the x term is 3/4, but I don't seem to be getting the other variables correct
okay, thanks

Answer 12

I think I made some error, so you may be correct. Let me recheck my work.

Answer 13

You are correct. \(x=3/4\)

Answer 14

Equation 5 should rather be\[3x+6z=0\]

Answer 15

Then, \[\begin{matrix}\quad \:\:13x-6z=12\\ \quad 3x+6z=0\\ \text{___________________}\\16x=12\end{matrix}\]So \(x=3/4\)

Answer 16

Then substituting into equation 5, \[3(3/4) +6z=0\]Means that \(z=-3/8\)

Answer 17

Substituting both of these into equation 2, we finally get that \[6(3/4)+y-(-3/8)=7\]Which means that \(y=17/8\).

Answer 18

Sorry about the confusion :(
I wrote down the wrong equations on my end.