Solve {l}{3x+2y+z=11}{5x+3y+4z=2}{x+y-z=1}

$$\left\{ \begin{array} { l } { 3 x + 2 y + z = 11 } \\ { 5 x + 3 y + 4 z = 2 } \\ { x + y - z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=66$$

$$y=-84$$

$$z=-19$$

Short Steps Using Substitution

Solve 3x+2y+z=11 for z.

$$z=-3x-2y+11$$

Substitute -3x-2y+11 for z in the second and third equation.

$$5x+3y+4\left(-3x-2y+11\right)=2 x+y-\left(-3x-2y+11\right)=1$$

Solve these equations for y and x respectively.

$$y=\frac{42}{5}-\frac{7}{5}x x=3-\frac{3}{4}y$$

Substitute \frac{42}{5}-\frac{7}{5}x for y in the equation x=3-\frac{3}{4}y.

$$x=3-\frac{3}{4}\left(\frac{42}{5}-\frac{7}{5}x\right)$$

Solve x=3-\frac{3}{4}\left(\frac{42}{5}-\frac{7}{5}x\right) for x.

$$x=66$$

Substitute 66 for x in the equation y=\frac{42}{5}-\frac{7}{5}x.

$$y=\frac{42}{5}-\frac{7}{5}\times 66$$

Calculate y from y=\frac{42}{5}-\frac{7}{5}\times 66.

$$y=-84$$

Substitute -84 for y and 66 for x in the equation z=-3x-2y+11.

$$z=-3\times 66-2\left(-84\right)+11$$

Calculate z from z=-3\times 66-2\left(-84\right)+11.

$$z=-19$$

The system is now solved.

$$x=66 y=-84 z=-19$$