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Sagot :
You have a system of 3 equations with 3 unknown variables.
[tex]\begin{gathered} 3x+2y+2z=-2\text{ (1)} \\ 2x+y-z=-2\text{ (2)} \\ x-3y+z=0\text{ (3)} \end{gathered}[/tex]You can start by applying the reduction or elimination method to solve the system. Let's add equation (3) and equation (2)
[tex]\begin{gathered} x-3y+z=0 \\ 2x+y-z=-2 \\ ------------ \\ 3x-2y+0=-2 \end{gathered}[/tex]This will be equation (4).
Now, let's apply the same method to equation (1) and (2) but, first you need to multiply equation (2) by 2:
[tex]\begin{gathered} 2(2x+y-z)=2(-2) \\ 4x+2y-2z=-4\text{ Eq. }(5) \end{gathered}[/tex]Add equation (1) and (5)
[tex]\begin{gathered} 3x+2y+2z=-2 \\ 4x+2y-2z=-4 \\ -------------- \\ 7x+4y+0=-6 \end{gathered}[/tex]This will be equation (6).
Now, let's apply the elimination method again to equations (4) and (6). But, first let's multiply equation (4) by 2:
[tex]\begin{gathered} 2(3x-2y)=2(-2) \\ 6x-4y=-4\text{ (7)} \end{gathered}[/tex]Now, let's add equations (6) and (7):
[tex]\begin{gathered} 7x+4y=-6 \\ 6x-4y=-4 \\ ----------- \\ 13x+0=-10 \\ \text{Now let's solve for x} \\ 13x=-10 \\ \text{Divide both sides by 13} \\ x=-\frac{10}{13} \end{gathered}[/tex]Now, you can replace this x-value into equation (6)
[tex]\begin{gathered} 7x+4y=-6 \\ 7(-\frac{10}{13})+4y=-6 \\ -\frac{70}{13}+4y=-6 \\ 4y=-6+\frac{70}{13} \\ 4y=\frac{-6\times13+70}{13}=\frac{-78+70}{13} \\ 4y=-\frac{8}{13} \\ \text{Divide both sides by 4} \\ \frac{4y}{4}=\frac{-\frac{8}{13}}{4} \\ \text{Simplify} \\ y=-\frac{8}{13\times4}=-\frac{2}{13} \end{gathered}[/tex]You know that x=-10/13, y=-2/13, now you can replace these values in any equation to find z-value:
Let's do it with equation (3)
[tex]\begin{gathered} x-3y+z=0 \\ -\frac{10}{13}-3(-\frac{2}{13})+z=0 \\ -\frac{10}{13}+\frac{6}{13}+z=0 \\ -\frac{4}{13}+z=0 \\ z=\frac{4}{13} \end{gathered}[/tex]The answer is X=-10/13, Y=-2/13 and Z=4/13
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