Solve {c}{2x+5y-z=1}{-3x-4y+z=-10}{4x+7y+2z=-5}

$$\left\{ \begin{array} { c } { 2 x + 5 y - z = 1 } \\ { - 3 x - 4 y + z = - 10 } \\ { 4 x + 7 y + 2 z = - 5 } \end{array} \right.$$

Solve for x, y, z

$$x=6$$

$$y=-3$$

$$z=-4$$

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.

$$-3x-4y+2x+5y-1=-10 4x+7y+2\left(2x+5y-1\right)=-5$$

Solve these equations for y and x respectively.

$$y=x-9 x=-\frac{17}{8}y-\frac{3}{8}$$

Substitute x-9 for y in the equation x=-\frac{17}{8}y-\frac{3}{8}.

$$x=-\frac{17}{8}\left(x-9\right)-\frac{3}{8}$$

Solve x=-\frac{17}{8}\left(x-9\right)-\frac{3}{8} for x.

$$x=6$$

Substitute 6 for x in the equation y=x-9.

$$y=6-9$$

Calculate y from y=6-9.

$$y=-3$$

Substitute -3 for y and 6 for x in the equation z=2x+5y-1.

$$z=2\times 6+5\left(-3\right)-1$$

Calculate z from z=2\times 6+5\left(-3\right)-1.

$$z=-4$$

The system is now solved.

$$x=6 y=-3 z=-4$$