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

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

Solve for x, y, z

$$x=-\frac{7}{13}\approx -0.538461538$$

$$y=\frac{9}{13}\approx 0.692307692$$

$$z = \frac{29}{13} = 2\frac{3}{13} \approx 2.230769231$$

Short Steps Using Substitution

Reorder the equations.

$$4x-3y+z=-2 2x+3y=1 x-y+z=1$$

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

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

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

$$x-y-4x+3y-2=1$$

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

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

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

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

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

$$x=-\frac{7}{13}$$

Substitute -\frac{7}{13} for x in the equation y=-\frac{2}{3}x+\frac{1}{3}.

$$y=-\frac{2}{3}\left(-\frac{7}{13}\right)+\frac{1}{3}$$

Calculate y from y=-\frac{2}{3}\left(-\frac{7}{13}\right)+\frac{1}{3}.

$$y=\frac{9}{13}$$

Substitute \frac{9}{13} for y and -\frac{7}{13} for x in the equation z=-4x+3y-2.

$$z=-4\left(-\frac{7}{13}\right)+3\times \frac{9}{13}-2$$

Calculate z from z=-4\left(-\frac{7}{13}\right)+3\times \frac{9}{13}-2.

$$z=\frac{29}{13}$$

The system is now solved.

$$x=-\frac{7}{13} y=\frac{9}{13} z=\frac{29}{13}$$