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

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

Solve for x, y, z

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$y = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

$$z = \frac{29}{6} = 4\frac{5}{6} \approx 4.833333333$$

Short Steps Using Substitution

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

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

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

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

Solve these equations for x and z respectively.

$$x=-\frac{3}{2} z=-\frac{17}{3}x-\frac{11}{3}$$

Substitute -\frac{3}{2} for x in the equation z=-\frac{17}{3}x-\frac{11}{3}.

$$z=-\frac{17}{3}\left(-\frac{3}{2}\right)-\frac{11}{3}$$

Calculate z from z=-\frac{17}{3}\left(-\frac{3}{2}\right)-\frac{11}{3}.

$$z=\frac{29}{6}$$

Substitute -\frac{3}{2} for x and \frac{29}{6} for z in the equation y=-3x-z-2.

$$y=-3\left(-\frac{3}{2}\right)-\frac{29}{6}-2$$

Calculate y from y=-3\left(-\frac{3}{2}\right)-\frac{29}{6}-2.

$$y=-\frac{7}{3}$$

The system is now solved.

$$x=-\frac{3}{2} y=-\frac{7}{3} z=\frac{29}{6}$$