Solve {l}{2x+6y+z=7}{x+2y-z=-1}{5x+7y-4z=9}

$$\left. \begin{array} { l } { 2 x + 6 y + z = 7 } \\ { x + 2 y - z = - 1 } \\ { 5 x + 7 y - 4 z = 9 } \end{array} \right.$$

Solve for x, y, z

$$x=10$$

$$y=-3$$

$$z=5$$

Short Steps Using Substitution

Solve 2x+6y+z=7 for z.

$$z=-2x-6y+7$$

Substitute -2x-6y+7 for z in the second and third equation.

$$x+2y-\left(-2x-6y+7\right)=-1 5x+7y-4\left(-2x-6y+7\right)=9$$

Solve these equations for y and x respectively.

$$y=\frac{3}{4}-\frac{3}{8}x x=-\frac{31}{13}y+\frac{37}{13}$$

Substitute \frac{3}{4}-\frac{3}{8}x for y in the equation x=-\frac{31}{13}y+\frac{37}{13}.

$$x=-\frac{31}{13}\left(\frac{3}{4}-\frac{3}{8}x\right)+\frac{37}{13}$$

Solve x=-\frac{31}{13}\left(\frac{3}{4}-\frac{3}{8}x\right)+\frac{37}{13} for x.

$$x=10$$

Substitute 10 for x in the equation y=\frac{3}{4}-\frac{3}{8}x.

$$y=\frac{3}{4}-\frac{3}{8}\times 10$$

Calculate y from y=\frac{3}{4}-\frac{3}{8}\times 10.

$$y=-3$$

Substitute -3 for y and 10 for x in the equation z=-2x-6y+7.

$$z=-2\times 10-6\left(-3\right)+7$$

Calculate z from z=-2\times 10-6\left(-3\right)+7.

$$z=5$$

The system is now solved.

$$x=10 y=-3 z=5$$