Solve {c}{2x+5y-z=1}{-x+y+5z=-13}{x+y-z=3}

$$\left\{ \begin{array} { c } { 2 x + 5 y - z = 1 } \\ { - x + y + 5 z = - 13 } \\ { x + y - z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=2$$

$$y=-1$$

$$z=-2$$

Short Steps Using Substitution

Solve 2x+5y-z=1 for z.

$$z=2x+5y-1$$

Substitute 2x+5y-1 for z in the second and third equation.

$$-x+y+5\left(2x+5y-1\right)=-13 x+y-\left(2x+5y-1\right)=3$$

Solve these equations for y and x respectively.

$$y=-\frac{9}{26}x-\frac{4}{13} x=-4y-2$$

Substitute -\frac{9}{26}x-\frac{4}{13} for y in the equation x=-4y-2.

$$x=-4\left(-\frac{9}{26}x-\frac{4}{13}\right)-2$$

Solve x=-4\left(-\frac{9}{26}x-\frac{4}{13}\right)-2 for x.

$$x=2$$

Substitute 2 for x in the equation y=-\frac{9}{26}x-\frac{4}{13}.

$$y=-\frac{9}{26}\times 2-\frac{4}{13}$$

Calculate y from y=-\frac{9}{26}\times 2-\frac{4}{13}.

$$y=-1$$

Substitute -1 for y and 2 for x in the equation z=2x+5y-1.

$$z=2\times 2+5\left(-1\right)-1$$

Calculate z from z=2\times 2+5\left(-1\right)-1.

$$z=-2$$

The system is now solved.

$$x=2 y=-1 z=-2$$