Solve {l}{2x+5y-z=8}{3x-2y+2z=3}{5x+3y+4z=9}

$$\left. \begin{array} { l } { 2 x + 5 y - z = 8 } \\ { 3 x - 2 y + 2 z = 3 } \\ { 5 x + 3 y + 4 z = 9 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{109}{57} = 1\frac{52}{57} \approx 1.912280702$$

$$y=\frac{40}{57}\approx 0.701754386$$

$$z=-\frac{2}{3}\approx -0.666666667$$

Short Steps Using Substitution

Solve 2x+5y-z=8 for z.

$$z=2x+5y-8$$

Substitute 2x+5y-8 for z in the second and third equation.

$$3x-2y+2\left(2x+5y-8\right)=3 5x+3y+4\left(2x+5y-8\right)=9$$

Solve these equations for y and x respectively.

$$y=-\frac{7}{8}x+\frac{19}{8} x=-\frac{23}{13}y+\frac{41}{13}$$

Substitute -\frac{7}{8}x+\frac{19}{8} for y in the equation x=-\frac{23}{13}y+\frac{41}{13}.

$$x=-\frac{23}{13}\left(-\frac{7}{8}x+\frac{19}{8}\right)+\frac{41}{13}$$

Solve x=-\frac{23}{13}\left(-\frac{7}{8}x+\frac{19}{8}\right)+\frac{41}{13} for x.

$$x=\frac{109}{57}$$

Substitute \frac{109}{57} for x in the equation y=-\frac{7}{8}x+\frac{19}{8}.

$$y=-\frac{7}{8}\times \frac{109}{57}+\frac{19}{8}$$

Calculate y from y=-\frac{7}{8}\times \frac{109}{57}+\frac{19}{8}.

$$y=\frac{40}{57}$$

Substitute \frac{40}{57} for y and \frac{109}{57} for x in the equation z=2x+5y-8.

$$z=2\times \frac{109}{57}+5\times \frac{40}{57}-8$$

Calculate z from z=2\times \frac{109}{57}+5\times \frac{40}{57}-8.

$$z=-\frac{2}{3}$$

The system is now solved.

$$x=\frac{109}{57} y=\frac{40}{57} z=-\frac{2}{3}$$