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

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

Solve for x, y, z

$$x=0$$

$$y=0$$

$$z=0$$

Short Steps Using Substitution

Solve -3x+4y+z=0 for z.

$$z=3x-4y$$

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

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

Solve these equations for y and x respectively.

$$y=\frac{5}{7}x x=\frac{21}{16}y$$

Substitute \frac{5}{7}x for y in the equation x=\frac{21}{16}y.

$$x=\frac{21}{16}\times \frac{5}{7}x$$

Solve x=\frac{21}{16}\times \frac{5}{7}x for x.

$$x=0$$

Substitute 0 for x in the equation y=\frac{5}{7}x.

$$y=\frac{5}{7}\times 0$$

Calculate y from y=\frac{5}{7}\times 0.

$$y=0$$

Substitute 0 for y and 0 for x in the equation z=3x-4y.

$$z=3\times 0-4\times 0$$

Calculate z from z=3\times 0-4\times 0.

$$z=0$$

The system is now solved.

$$x=0 y=0 z=0$$