Solve {l}{2x+5y=0}{3x+6y=1}{x+z+y=-2}

$$\left. \begin{array} { l } { 2 x + 5 y = 0 } \\ { 3 x + 6 y = 1 } \\ { x + z + y = - 2 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$y=-\frac{2}{3}\approx -0.666666667$$

$$z=-3$$

Short Steps Using Substitution

Reorder the equations.

$$x+z+y=-2 3x+6y=1 2x+5y=0$$

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

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

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

$$3\left(-z-y-2\right)+6y=1 2\left(-z-y-2\right)+5y=0$$

Solve these equations for y and z respectively.

$$y=z+\frac{7}{3} z=-2+\frac{3}{2}y$$

Substitute z+\frac{7}{3} for y in the equation z=-2+\frac{3}{2}y.

$$z=-2+\frac{3}{2}\left(z+\frac{7}{3}\right)$$

Solve z=-2+\frac{3}{2}\left(z+\frac{7}{3}\right) for z.

$$z=-3$$

Substitute -3 for z in the equation y=z+\frac{7}{3}.

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

Calculate y from y=-3+\frac{7}{3}.

$$y=-\frac{2}{3}$$

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

$$x=-\left(-3\right)-\left(-\frac{2}{3}\right)-2$$

Calculate x from x=-\left(-3\right)-\left(-\frac{2}{3}\right)-2.

$$x=\frac{5}{3}$$

The system is now solved.

$$x=\frac{5}{3} y=-\frac{2}{3} z=-3$$