Solve {c}{x+y-3z=-36}{2x-y+6z=78}{x-2y+5z=42}

$$\left. \begin{array} { c } { x + y - 3 z = - 36 } \\ { 2 x - y + 6 z = 78 } \\ { x - 2 y + 5 z = 42 } \end{array} \right.$$

Solve for x, y, z

$$x=-4$$

$$y=22$$

$$z=18$$

Short Steps Using Substitution

Solve x+y-3z=-36 for x.

$$x=-y+3z-36$$

Substitute -y+3z-36 for x in the second and third equation.

$$2\left(-y+3z-36\right)-y+6z=78 -y+3z-36-2y+5z=42$$

Solve these equations for y and z respectively.

$$y=-50+4z z=\frac{39}{4}+\frac{3}{8}y$$

Substitute -50+4z for y in the equation z=\frac{39}{4}+\frac{3}{8}y.

$$z=\frac{39}{4}+\frac{3}{8}\left(-50+4z\right)$$

Solve z=\frac{39}{4}+\frac{3}{8}\left(-50+4z\right) for z.

$$z=18$$

Substitute 18 for z in the equation y=-50+4z.

$$y=-50+4\times 18$$

Calculate y from y=-50+4\times 18.

$$y=22$$

Substitute 22 for y and 18 for z in the equation x=-y+3z-36.

$$x=-22+3\times 18-36$$

Calculate x from x=-22+3\times 18-36.

$$x=-4$$

The system is now solved.

$$x=-4 y=22 z=18$$