Solve {l}{x+2y+3z=1}{2x-z=5}{5y+5z=25}

$$\left. \begin{array} { l } { x + 2 y + 3 z = 1 } \\ { 2 x - z = 5 } \\ { 5 y + 5 z = 25 } \end{array} \right.$$

Solve for x, y, z

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

$$y = \frac{38}{3} = 12\frac{2}{3} \approx 12.666666667$$

$$z = -\frac{23}{3} = -7\frac{2}{3} \approx -7.666666667$$

Short Steps Using Substitution

Solve x+2y+3z=1 for x.

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

Substitute -2y-3z+1 for x in the equation 2x-z=5.

$$2\left(-2y-3z+1\right)-z=5$$

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

$$y=-\frac{3}{4}-\frac{7}{4}z z=5-y$$

Substitute -\frac{3}{4}-\frac{7}{4}z for y in the equation z=5-y.

$$z=5-\left(-\frac{3}{4}-\frac{7}{4}z\right)$$

Solve z=5-\left(-\frac{3}{4}-\frac{7}{4}z\right) for z.

$$z=-\frac{23}{3}$$

Substitute -\frac{23}{3} for z in the equation y=-\frac{3}{4}-\frac{7}{4}z.

$$y=-\frac{3}{4}-\frac{7}{4}\left(-\frac{23}{3}\right)$$

Calculate y from y=-\frac{3}{4}-\frac{7}{4}\left(-\frac{23}{3}\right).

$$y=\frac{38}{3}$$

Substitute \frac{38}{3} for y and -\frac{23}{3} for z in the equation x=-2y-3z+1.

$$x=-2\times \frac{38}{3}-3\left(-\frac{23}{3}\right)+1$$

Calculate x from x=-2\times \frac{38}{3}-3\left(-\frac{23}{3}\right)+1.

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

The system is now solved.

$$x=-\frac{4}{3} y=\frac{38}{3} z=-\frac{23}{3}$$