Solve {l}{x+y+z=23}{x-y=1}{2x+y-2=20}

$$\left\{ \begin{array} { l } { x + y + z = 23 } \\ { x - y = 1 } \\ { 2 x + y - 2 = 20 } \end{array} \right.$$

Solve for x, y, z

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

$$y = \frac{20}{3} = 6\frac{2}{3} \approx 6.666666667$$

$$z = \frac{26}{3} = 8\frac{2}{3} \approx 8.666666667$$

Short Steps Using Substitution

Solve x+y+z=23 for x.

$$x=-y-z+23$$

Substitute -y-z+23 for x in the second and third equation.

$$-y-z+23-y=1 2\left(-y-z+23\right)+y-2=20$$

Solve these equations for y and z respectively.

$$y=-\frac{1}{2}z+11 z=12-\frac{1}{2}y$$

Substitute -\frac{1}{2}z+11 for y in the equation z=12-\frac{1}{2}y.

$$z=12-\frac{1}{2}\left(-\frac{1}{2}z+11\right)$$

Solve z=12-\frac{1}{2}\left(-\frac{1}{2}z+11\right) for z.

$$z=\frac{26}{3}$$

Substitute \frac{26}{3} for z in the equation y=-\frac{1}{2}z+11.

$$y=-\frac{1}{2}\times \frac{26}{3}+11$$

Calculate y from y=-\frac{1}{2}\times \frac{26}{3}+11.

$$y=\frac{20}{3}$$

Substitute \frac{20}{3} for y and \frac{26}{3} for z in the equation x=-y-z+23.

$$x=-\frac{20}{3}-\frac{26}{3}+23$$

Calculate x from x=-\frac{20}{3}-\frac{26}{3}+23.

$$x=\frac{23}{3}$$

The system is now solved.

$$x=\frac{23}{3} y=\frac{20}{3} z=\frac{26}{3}$$