Solve {c}{x+3y-2z=12}{2x+3y-4y=18}{4x+y+2z=11}

$$\left. \begin{array} { c } { x + 3 y - 2 z = 12 } \\ { 2 x + 3 y - 4 y = 18 } \\ { 4 x + y + 2 z = 11 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{95}{13} = 7\frac{4}{13} \approx 7.307692308$$

$$y = -\frac{44}{13} = -3\frac{5}{13} \approx -3.384615385$$

$$z = -\frac{193}{26} = -7\frac{11}{26} \approx -7.423076923$$

Short Steps Using Substitution

Solve x+3y-2z=12 for x.

$$x=-3y+2z+12$$

Substitute -3y+2z+12 for x in the second and third equation.

$$2\left(-3y+2z+12\right)+3y-4y=18 4\left(-3y+2z+12\right)+y+2z=11$$

Solve these equations for y and z respectively.

$$y=\frac{6}{7}+\frac{4}{7}z z=-\frac{37}{10}+\frac{11}{10}y$$

Substitute \frac{6}{7}+\frac{4}{7}z for y in the equation z=-\frac{37}{10}+\frac{11}{10}y.

$$z=-\frac{37}{10}+\frac{11}{10}\left(\frac{6}{7}+\frac{4}{7}z\right)$$

Solve z=-\frac{37}{10}+\frac{11}{10}\left(\frac{6}{7}+\frac{4}{7}z\right) for z.

$$z=-\frac{193}{26}$$

Substitute -\frac{193}{26} for z in the equation y=\frac{6}{7}+\frac{4}{7}z.

$$y=\frac{6}{7}+\frac{4}{7}\left(-\frac{193}{26}\right)$$

Calculate y from y=\frac{6}{7}+\frac{4}{7}\left(-\frac{193}{26}\right).

$$y=-\frac{44}{13}$$

Substitute -\frac{44}{13} for y and -\frac{193}{26} for z in the equation x=-3y+2z+12.

$$x=-3\left(-\frac{44}{13}\right)+2\left(-\frac{193}{26}\right)+12$$

Calculate x from x=-3\left(-\frac{44}{13}\right)+2\left(-\frac{193}{26}\right)+12.

$$x=\frac{95}{13}$$

The system is now solved.

$$x=\frac{95}{13} y=-\frac{44}{13} z=-\frac{193}{26}$$