\left\{ \begin{array} { l } { x + 2 y - 3 z = 12 } \\ { 3 x + y - 2 z = 5 } \\ { 2 x - 3 y - 5 z = 7 } \end{array} \right.
Solve for x, y, z
x=-1
y=2
z=-3
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x=-2y+3z+12
Solve x+2y-3z=12 for x.
3\left(-2y+3z+12\right)+y-2z=5 2\left(-2y+3z+12\right)-3y-5z=7
Substitute -2y+3z+12 for x in the second and third equation.
y=\frac{7}{5}z+\frac{31}{5} z=7y-17
Solve these equations for y and z respectively.
z=7\left(\frac{7}{5}z+\frac{31}{5}\right)-17
Substitute \frac{7}{5}z+\frac{31}{5} for y in the equation z=7y-17.
z=-3
Solve z=7\left(\frac{7}{5}z+\frac{31}{5}\right)-17 for z.
y=\frac{7}{5}\left(-3\right)+\frac{31}{5}
Substitute -3 for z in the equation y=\frac{7}{5}z+\frac{31}{5}.
y=2
Calculate y from y=\frac{7}{5}\left(-3\right)+\frac{31}{5}.
x=-2\times 2+3\left(-3\right)+12
Substitute 2 for y and -3 for z in the equation x=-2y+3z+12.
x=-1
Calculate x from x=-2\times 2+3\left(-3\right)+12.
x=-1 y=2 z=-3
The system is now solved.
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