Solve {l}{2x+y=7}{2y+z=8}{2z+x=9}

$$\left\{ \begin{array} { l } { 2 x + y = 7 } \\ { 2 y + z = 8 } \\ { 2 z + x = 9 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

$$y = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

$$z = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333$$

Short Steps Using Substitution

Solve 2x+y=7 for y.

$$y=-2x+7$$

Substitute -2x+7 for y in the equation 2y+z=8.

$$2\left(-2x+7\right)+z=8$$

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

$$x=\frac{3}{2}+\frac{1}{4}z z=-\frac{1}{2}x+\frac{9}{2}$$

Substitute \frac{3}{2}+\frac{1}{4}z for x in the equation z=-\frac{1}{2}x+\frac{9}{2}.

$$z=-\frac{1}{2}\left(\frac{3}{2}+\frac{1}{4}z\right)+\frac{9}{2}$$

Solve z=-\frac{1}{2}\left(\frac{3}{2}+\frac{1}{4}z\right)+\frac{9}{2} for z.

$$z=\frac{10}{3}$$

Substitute \frac{10}{3} for z in the equation x=\frac{3}{2}+\frac{1}{4}z.

$$x=\frac{3}{2}+\frac{1}{4}\times \frac{10}{3}$$

Calculate x from x=\frac{3}{2}+\frac{1}{4}\times \frac{10}{3}.

$$x=\frac{7}{3}$$

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

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

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

$$y=\frac{7}{3}$$

The system is now solved.

$$x=\frac{7}{3} y=\frac{7}{3} z=\frac{10}{3}$$