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

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

Solve for x, y, z

$$x=\frac{13}{22}\approx 0.590909091$$

$$y=\frac{5}{22}\approx 0.227272727$$

$$z=\frac{17}{22}\approx 0.772727273$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for y and x respectively.

$$y=-x+\frac{9}{11} x=-\frac{1}{3}y+\frac{2}{3}$$

Substitute -x+\frac{9}{11} for y in the equation x=-\frac{1}{3}y+\frac{2}{3}.

$$x=-\frac{1}{3}\left(-x+\frac{9}{11}\right)+\frac{2}{3}$$

Solve x=-\frac{1}{3}\left(-x+\frac{9}{11}\right)+\frac{2}{3} for x.

$$x=\frac{13}{22}$$

Substitute \frac{13}{22} for x in the equation y=-x+\frac{9}{11}.

$$y=-\frac{13}{22}+\frac{9}{11}$$

Calculate y from y=-\frac{13}{22}+\frac{9}{11}.

$$y=\frac{5}{22}$$

Substitute \frac{5}{22} for y and \frac{13}{22} for x in the equation z=-3x-2y+3.

$$z=-3\times \frac{13}{22}-2\times \frac{5}{22}+3$$

Calculate z from z=-3\times \frac{13}{22}-2\times \frac{5}{22}+3.

$$z=\frac{17}{22}$$

The system is now solved.

$$x=\frac{13}{22} y=\frac{5}{22} z=\frac{17}{22}$$