Solve {l}{4x-2y-z=-3}{-x+y+2z=7}{2x-5y+z=-5}

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

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=3$$

Short Steps Using Substitution

Solve 4x-2y-z=-3 for z.

$$z=4x-2y+3$$

Substitute 4x-2y+3 for z in the second and third equation.

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

Solve these equations for y and x respectively.

$$y=\frac{7}{3}x-\frac{1}{3} x=\frac{7}{6}y-\frac{4}{3}$$

Substitute \frac{7}{3}x-\frac{1}{3} for y in the equation x=\frac{7}{6}y-\frac{4}{3}.

$$x=\frac{7}{6}\left(\frac{7}{3}x-\frac{1}{3}\right)-\frac{4}{3}$$

Solve x=\frac{7}{6}\left(\frac{7}{3}x-\frac{1}{3}\right)-\frac{4}{3} for x.

$$x=1$$

Substitute 1 for x in the equation y=\frac{7}{3}x-\frac{1}{3}.

$$y=\frac{7}{3}\times 1-\frac{1}{3}$$

Calculate y from y=\frac{7}{3}\times 1-\frac{1}{3}.

$$y=2$$

Substitute 2 for y and 1 for x in the equation z=4x-2y+3.

$$z=4\times 1-2\times 2+3$$

Calculate z from z=4\times 1-2\times 2+3.

$$z=3$$

The system is now solved.

$$x=1 y=2 z=3$$