Solve {l}{3x-2y+z=0}{x-y+z=0}{4x+2y-3z=5}

$$\left. \begin{array} { l } { 3 x - 2 y + z = 0 } \\ { x - y + z = 0 } \\ { 4 x + 2 y - 3 z = 5 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=1$$

Short Steps Using Substitution

Solve 3x-2y+z=0 for z.

$$z=-3x+2y$$

Substitute -3x+2y for z in the second and third equation.

$$x-y-3x+2y=0 4x+2y-3\left(-3x+2y\right)=5$$

Solve these equations for y and x respectively.

$$y=2x x=\frac{4}{13}y+\frac{5}{13}$$

Substitute 2x for y in the equation x=\frac{4}{13}y+\frac{5}{13}.

$$x=\frac{4}{13}\times 2x+\frac{5}{13}$$

Solve x=\frac{4}{13}\times 2x+\frac{5}{13} for x.

$$x=1$$

Substitute 1 for x in the equation y=2x.

$$y=2\times 1$$

Calculate y from y=2\times 1.

$$y=2$$

Substitute 2 for y and 1 for x in the equation z=-3x+2y.

$$z=-3+2\times 2$$

Calculate z from z=-3+2\times 2.

$$z=1$$

The system is now solved.

$$x=1 y=2 z=1$$