Solve {l}{5x-2y+z=14}{3x+y+z=4}{3x+y-z=12}

$$\left. \begin{array} { l } { 5 x - 2 y + z = 14 } \\ { 3 x + y + z = 4 } \\ { 3 x + y - z = 12 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{34}{11} = 3\frac{1}{11} \approx 3.090909091$$

$$y = -\frac{14}{11} = -1\frac{3}{11} \approx -1.272727273$$

$$z=-4$$

Short Steps Using Substitution

Solve 5x-2y+z=14 for z.

$$z=-5x+2y+14$$

Substitute -5x+2y+14 for z in the second and third equation.

$$3x+y-5x+2y+14=4 3x+y-\left(-5x+2y+14\right)=12$$

Solve these equations for y and x respectively.

$$y=-\frac{10}{3}+\frac{2}{3}x x=\frac{13}{4}+\frac{1}{8}y$$

Substitute -\frac{10}{3}+\frac{2}{3}x for y in the equation x=\frac{13}{4}+\frac{1}{8}y.

$$x=\frac{13}{4}+\frac{1}{8}\left(-\frac{10}{3}+\frac{2}{3}x\right)$$

Solve x=\frac{13}{4}+\frac{1}{8}\left(-\frac{10}{3}+\frac{2}{3}x\right) for x.

$$x=\frac{34}{11}$$

Substitute \frac{34}{11} for x in the equation y=-\frac{10}{3}+\frac{2}{3}x.

$$y=-\frac{10}{3}+\frac{2}{3}\times \frac{34}{11}$$

Calculate y from y=-\frac{10}{3}+\frac{2}{3}\times \frac{34}{11}.

$$y=-\frac{14}{11}$$

Substitute -\frac{14}{11} for y and \frac{34}{11} for x in the equation z=-5x+2y+14.

$$z=-5\times \frac{34}{11}+2\left(-\frac{14}{11}\right)+14$$

Calculate z from z=-5\times \frac{34}{11}+2\left(-\frac{14}{11}\right)+14.

$$z=-4$$

The system is now solved.

$$x=\frac{34}{11} y=-\frac{14}{11} z=-4$$