Solve {r}{4x-3y+z=-20}{-2x+y-3z=-8}{x-y+2z=7}

$$\left\{ \begin{array} { r } { 4 x - 3 y + z = - 20 } \\ { - 2 x + y - 3 z = - 8 } \\ { x - y + 2 z = 7 } \end{array} \right.$$

Solve for x, y, z

$$x=-6$$

$$y=1$$

$$z=7$$

Short Steps Using Substitution

Solve 4x-3y+z=-20 for z.

$$z=-4x+3y-20$$

Substitute -4x+3y-20 for z in the second and third equation.

$$-2x+y-3\left(-4x+3y-20\right)=-8 x-y+2\left(-4x+3y-20\right)=7$$

Solve these equations for y and x respectively.

$$y=\frac{17}{2}+\frac{5}{4}x x=\frac{5}{7}y-\frac{47}{7}$$

Substitute \frac{17}{2}+\frac{5}{4}x for y in the equation x=\frac{5}{7}y-\frac{47}{7}.

$$x=\frac{5}{7}\left(\frac{17}{2}+\frac{5}{4}x\right)-\frac{47}{7}$$

Solve x=\frac{5}{7}\left(\frac{17}{2}+\frac{5}{4}x\right)-\frac{47}{7} for x.

$$x=-6$$

Substitute -6 for x in the equation y=\frac{17}{2}+\frac{5}{4}x.

$$y=\frac{17}{2}+\frac{5}{4}\left(-6\right)$$

Calculate y from y=\frac{17}{2}+\frac{5}{4}\left(-6\right).

$$y=1$$

Substitute 1 for y and -6 for x in the equation z=-4x+3y-20.

$$z=-4\left(-6\right)+3\times 1-20$$

Calculate z from z=-4\left(-6\right)+3\times 1-20.

$$z=7$$

The system is now solved.

$$x=-6 y=1 z=7$$