Solve {l}{4x-3y+5z=32}{5x+2y+z=11}{7x-y+3z=14}

$$\left\{ \begin{array} { l } { 4 x - 3 y + 5 z = 32 } \\ { 5 x + 2 y + z = 11 } \\ { 7 x - y + 3 z = 14 } \end{array} \right.$$

Solve for x, y, z

$$x=-2$$

$$y=5$$

$$z=11$$

Short Steps Using Substitution

Reorder the equations.

$$5x+2y+z=11 4x-3y+5z=32 7x-y+3z=14$$

Solve 5x+2y+z=11 for z.

$$z=-5x-2y+11$$

Substitute -5x-2y+11 for z in the second and third equation.

$$4x-3y+5\left(-5x-2y+11\right)=32 7x-y+3\left(-5x-2y+11\right)=14$$

Solve these equations for y and x respectively.

$$y=-\frac{21}{13}x+\frac{23}{13} x=-\frac{7}{8}y+\frac{19}{8}$$

Substitute -\frac{21}{13}x+\frac{23}{13} for y in the equation x=-\frac{7}{8}y+\frac{19}{8}.

$$x=-\frac{7}{8}\left(-\frac{21}{13}x+\frac{23}{13}\right)+\frac{19}{8}$$

Solve x=-\frac{7}{8}\left(-\frac{21}{13}x+\frac{23}{13}\right)+\frac{19}{8} for x.

$$x=-2$$

Substitute -2 for x in the equation y=-\frac{21}{13}x+\frac{23}{13}.

$$y=-\frac{21}{13}\left(-2\right)+\frac{23}{13}$$

Calculate y from y=-\frac{21}{13}\left(-2\right)+\frac{23}{13}.

$$y=5$$

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

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

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

$$z=11$$

The system is now solved.

$$x=-2 y=5 z=11$$