Solve {r}{x-y+z=7}{2x-z=0}{2y+3z=9}

$$\left\{ \begin{array} { r } { x - y + z = 7 } \\ { 2 x - z = 0 } \\ { 2 y + 3 z = 9 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{23}{12} = 1\frac{11}{12} \approx 1.916666667$$

$$y = -\frac{5}{4} = -1\frac{1}{4} = -1.25$$

$$z = \frac{23}{6} = 3\frac{5}{6} \approx 3.833333333$$

Short Steps Using Substitution

Solve x-y+z=7 for x.

$$x=y-z+7$$

Substitute y-z+7 for x in the equation 2x-z=0.

$$2\left(y-z+7\right)-z=0$$

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

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

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

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

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

$$z=\frac{23}{6}$$

Substitute \frac{23}{6} for z in the equation y=-7+\frac{3}{2}z.

$$y=-7+\frac{3}{2}\times \frac{23}{6}$$

Calculate y from y=-7+\frac{3}{2}\times \frac{23}{6}.

$$y=-\frac{5}{4}$$

Substitute -\frac{5}{4} for y and \frac{23}{6} for z in the equation x=y-z+7.

$$x=-\frac{5}{4}-\frac{23}{6}+7$$

Calculate x from x=-\frac{5}{4}-\frac{23}{6}+7.

$$x=\frac{23}{12}$$

The system is now solved.

$$x=\frac{23}{12} y=-\frac{5}{4} z=\frac{23}{6}$$