Solve {l}{5x+3y-z=6}{4x+5y-10z=4}{2x+8y-3z=2}

$$\left. \begin{array}{l}{ 5 x + 3 y - z = 6 }\\{ 4 x + 5 y - 10 z = 4 }\\{ 2 x + 8 y - 3 z = 2 }\end{array} \right.$$

Solve for x, y, z

$$x = \frac{344}{279} = 1\frac{65}{279} \approx 1.23297491$$

$$y=-\frac{8}{279}\approx -0.028673835$$

$$z=\frac{22}{279}\approx 0.078853047$$

Short Steps Using Substitution

Solve 5x+3y-z=6 for z.

$$z=5x+3y-6$$

Substitute 5x+3y-6 for z in the second and third equation.

$$4x+5y-10\left(5x+3y-6\right)=4 2x+8y-3\left(5x+3y-6\right)=2$$

Solve these equations for y and x respectively.

$$y=\frac{56}{25}-\frac{46}{25}x x=-\frac{1}{13}y+\frac{16}{13}$$

Substitute \frac{56}{25}-\frac{46}{25}x for y in the equation x=-\frac{1}{13}y+\frac{16}{13}.

$$x=-\frac{1}{13}\left(\frac{56}{25}-\frac{46}{25}x\right)+\frac{16}{13}$$

Solve x=-\frac{1}{13}\left(\frac{56}{25}-\frac{46}{25}x\right)+\frac{16}{13} for x.

$$x=\frac{344}{279}$$

Substitute \frac{344}{279} for x in the equation y=\frac{56}{25}-\frac{46}{25}x.

$$y=\frac{56}{25}-\frac{46}{25}\times \frac{344}{279}$$

Calculate y from y=\frac{56}{25}-\frac{46}{25}\times \frac{344}{279}.

$$y=-\frac{8}{279}$$

Substitute -\frac{8}{279} for y and \frac{344}{279} for x in the equation z=5x+3y-6.

$$z=5\times \frac{344}{279}+3\left(-\frac{8}{279}\right)-6$$

Calculate z from z=5\times \frac{344}{279}+3\left(-\frac{8}{279}\right)-6.

$$z=\frac{22}{279}$$

The system is now solved.

$$x=\frac{344}{279} y=-\frac{8}{279} z=\frac{22}{279}$$