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

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

Solve for x, y, z

$$x = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333$$

$$y = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

$$z = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Short Steps Using Substitution

Solve x+y+3z=5 for x.

$$x=-y-3z+5$$

Substitute -y-3z+5 for x in the second and third equation.

$$-y-3z+5+2z=2 3\left(-y-3z+5\right)+2y+2z=2$$

Solve these equations for y and z respectively.

$$y=-z+3 z=-\frac{1}{7}y+\frac{13}{7}$$

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

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

Solve z=-\frac{1}{7}\left(-z+3\right)+\frac{13}{7} for z.

$$z=\frac{5}{3}$$

Substitute \frac{5}{3} for z in the equation y=-z+3.

$$y=-\frac{5}{3}+3$$

Calculate y from y=-\frac{5}{3}+3.

$$y=\frac{4}{3}$$

Substitute \frac{4}{3} for y and \frac{5}{3} for z in the equation x=-y-3z+5.

$$x=-\frac{4}{3}-3\times \frac{5}{3}+5$$

Calculate x from x=-\frac{4}{3}-3\times \frac{5}{3}+5.

$$x=-\frac{4}{3}$$

The system is now solved.

$$x=-\frac{4}{3} y=\frac{4}{3} z=\frac{5}{3}$$