Question

Solve the following system of equations for all three variables. 2x – y + 10z = 3x + y + 6z = -2X – Y + 2z = 0

Answer 1

Let:

$\begin{gathered} 2x-y+10z=3_{\text{ }}(1) \\ x+y+6z=-2_{\text{ }}(2) \\ x-y+2z=0_{\text{ }}(3) \end{gathered}$

Using elimination:

$\begin{gathered} (2)+(3) \\ x+x+y-y+6z+2z=-2+0 \\ 2x+8z=-2_{\text{ }}(4) \end{gathered}$$\begin{gathered} (1)-(3) \\ 2x-x-y-(-y)+10z-2z=3-0 \\ x+8z=3_{\text{ }}(5) \end{gathered}$$\begin{gathered} (4)-(5) \\ 2x-x+8z-8z=-2-3 \\ x=-5 \end{gathered}$

Replace the value of x into (5):

$\begin{gathered} -5+8z=3 \\ 8z=3+5 \\ 8z=8 \\ z=\frac{8}{8} \\ z=1 \end{gathered}$

Replace z and x into (3):

$\begin{gathered} -5-y+2(1)=0 \\ y=-5+2 \\ y=-3 \end{gathered}$