Solve {l}{x+2y-z=4}{4x-3y+2=8}{5x-y=12}

$$\left. \begin{array} { l } { x + 2 y - z = 4 } \\ { 4 x - 3 y + 2 = 8 } \\ { 5 x - y = 12 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{30}{11} = 2\frac{8}{11} \approx 2.727272727$$

$$y = \frac{18}{11} = 1\frac{7}{11} \approx 1.636363636$$

$$z=2$$

Short Steps Using Substitution

Solve x+2y-z=4 for x.

$$x=-2y+z+4$$

Substitute -2y+z+4 for x in the second and third equation.

$$4\left(-2y+z+4\right)-3y+2=8 5\left(-2y+z+4\right)-y=12$$

Solve these equations for y and z respectively.

$$y=\frac{10}{11}+\frac{4}{11}z z=-\frac{8}{5}+\frac{11}{5}y$$

Substitute \frac{10}{11}+\frac{4}{11}z for y in the equation z=-\frac{8}{5}+\frac{11}{5}y.

$$z=-\frac{8}{5}+\frac{11}{5}\left(\frac{10}{11}+\frac{4}{11}z\right)$$

Solve z=-\frac{8}{5}+\frac{11}{5}\left(\frac{10}{11}+\frac{4}{11}z\right) for z.

$$z=2$$

Substitute 2 for z in the equation y=\frac{10}{11}+\frac{4}{11}z.

$$y=\frac{10}{11}+\frac{4}{11}\times 2$$

Calculate y from y=\frac{10}{11}+\frac{4}{11}\times 2.

$$y=\frac{18}{11}$$

Substitute \frac{18}{11} for y and 2 for z in the equation x=-2y+z+4.

$$x=-2\times \frac{18}{11}+2+4$$

Calculate x from x=-2\times \frac{18}{11}+2+4.

$$x=\frac{30}{11}$$

The system is now solved.

$$x=\frac{30}{11} y=\frac{18}{11} z=2$$