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

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

Solve for x, y, z

$$x=-11$$

$$y=16$$

$$z=5$$

Short Steps Using Substitution

Reorder the equations.

$$5x+3y+z=-2 2x+2y-2z=0 4x+3y=4$$

Solve 5x+3y+z=-2 for z.

$$z=-5x-3y-2$$

Substitute -5x-3y-2 for z in the equation 2x+2y-2z=0.

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

Solve the second equation for y and the third equation for x.

$$y=-\frac{1}{2}-\frac{3}{2}x x=-\frac{3}{4}y+1$$

Substitute -\frac{1}{2}-\frac{3}{2}x for y in the equation x=-\frac{3}{4}y+1.

$$x=-\frac{3}{4}\left(-\frac{1}{2}-\frac{3}{2}x\right)+1$$

Solve x=-\frac{3}{4}\left(-\frac{1}{2}-\frac{3}{2}x\right)+1 for x.

$$x=-11$$

Substitute -11 for x in the equation y=-\frac{1}{2}-\frac{3}{2}x.

$$y=-\frac{1}{2}-\frac{3}{2}\left(-11\right)$$

Calculate y from y=-\frac{1}{2}-\frac{3}{2}\left(-11\right).

$$y=16$$

Substitute 16 for y and -11 for x in the equation z=-5x-3y-2.

$$z=-5\left(-11\right)-3\times 16-2$$

Calculate z from z=-5\left(-11\right)-3\times 16-2.

$$z=5$$

The system is now solved.

$$x=-11 y=16 z=5$$