Solve {c}{5x+y+4z=14}{-3x+2y-5z=2}{x-y+3z=-6}

$$\left\{ \begin{array} { c } { 5 x + y + 4 z = 14 } \\ { - 3 x + 2 y - 5 z = 2 } \\ { x - y + 3 z = - 6 } \end{array} \right.$$

Solve for x, y, z

$$x=6$$

$$y=0$$

$$z=-4$$

Short Steps Using Substitution

Solve 5x+y+4z=14 for y.

$$y=-5x-4z+14$$

Substitute -5x-4z+14 for y in the second and third equation.

$$-3x+2\left(-5x-4z+14\right)-5z=2 x-\left(-5x-4z+14\right)+3z=-6$$

Solve these equations for x and z respectively.

$$x=2-z z=\frac{8}{7}-\frac{6}{7}x$$

Substitute 2-z for x in the equation z=\frac{8}{7}-\frac{6}{7}x.

$$z=\frac{8}{7}-\frac{6}{7}\left(2-z\right)$$

Solve z=\frac{8}{7}-\frac{6}{7}\left(2-z\right) for z.

$$z=-4$$

Substitute -4 for z in the equation x=2-z.

$$x=2-\left(-4\right)$$

Calculate x from x=2-\left(-4\right).

$$x=6$$

Substitute 6 for x and -4 for z in the equation y=-5x-4z+14.

$$y=-5\times 6-4\left(-4\right)+14$$

Calculate y from y=-5\times 6-4\left(-4\right)+14.

$$y=0$$

The system is now solved.

$$x=6 y=0 z=-4$$