Solve {l}{3x+2y-z=1}{2x-2y+4z=-1}{-x+1/2y-x=0}

$$\left\{ \begin{array}{l}{ 3 x + 2 y - z = 1 }\\{ 2 x - 2 y + 4 z = - 1 }\\{ - x + \frac { 1 } { 2 } y - x = 0 }\end{array} \right.$$

Solve for x, y, z

$$x=\frac{3}{38}\approx 0.078947368$$

$$y=\frac{6}{19}\approx 0.315789474$$

$$z=-\frac{5}{38}\approx -0.131578947$$

Short Steps Using Substitution

Solve 3x+2y-z=1 for z.

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

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

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

Solve the second equation for y and the third equation for x.

$$y=-\frac{7}{3}x+\frac{1}{2} x=\frac{1}{4}y$$

Substitute -\frac{7}{3}x+\frac{1}{2} for y in the equation x=\frac{1}{4}y.

$$x=\frac{1}{4}\left(-\frac{7}{3}x+\frac{1}{2}\right)$$

Solve x=\frac{1}{4}\left(-\frac{7}{3}x+\frac{1}{2}\right) for x.

$$x=\frac{3}{38}$$

Substitute \frac{3}{38} for x in the equation y=-\frac{7}{3}x+\frac{1}{2}.

$$y=-\frac{7}{3}\times \frac{3}{38}+\frac{1}{2}$$

Calculate y from y=-\frac{7}{3}\times \frac{3}{38}+\frac{1}{2}.

$$y=\frac{6}{19}$$

Substitute \frac{6}{19} for y and \frac{3}{38} for x in the equation z=3x+2y-1.

$$z=3\times \frac{3}{38}+2\times \frac{6}{19}-1$$

Calculate z from z=3\times \frac{3}{38}+2\times \frac{6}{19}-1.

$$z=-\frac{5}{38}$$

The system is now solved.

$$x=\frac{3}{38} y=\frac{6}{19} z=-\frac{5}{38}$$