Solve {l}{3x+5y=z}{2x+6z=9}{3x+2z=y}

$$\left. \begin{array}{l}{ 3 x + 5 y = z }\\{ 2 x + 6 z = 9 }\\{ 3 x + 2 z = y }\end{array} \right.$$

Solve for x, y, z

$$x=-\frac{9}{10}=-0.9$$

$$y=\frac{9}{10}=0.9$$

$$z = \frac{9}{5} = 1\frac{4}{5} = 1.8$$

Short Steps Using Substitution

Solve 3x+5y=z for z.

$$z=3x+5y$$

Substitute 3x+5y for z in the second and third equation.

$$2x+6\left(3x+5y\right)=9 3x+2\left(3x+5y\right)=y$$

Solve these equations for y and x respectively.

$$y=\frac{3}{10}-\frac{2}{3}x x=-y$$

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

$$x=-\left(\frac{3}{10}-\frac{2}{3}x\right)$$

Solve x=-\left(\frac{3}{10}-\frac{2}{3}x\right) for x.

$$x=-\frac{9}{10}$$

Substitute -\frac{9}{10} for x in the equation y=\frac{3}{10}-\frac{2}{3}x.

$$y=\frac{3}{10}-\frac{2}{3}\left(-\frac{9}{10}\right)$$

Calculate y from y=\frac{3}{10}-\frac{2}{3}\left(-\frac{9}{10}\right).

$$y=\frac{9}{10}$$

Substitute \frac{9}{10} for y and -\frac{9}{10} for x in the equation z=3x+5y.

$$z=3\left(-\frac{9}{10}\right)+5\times \frac{9}{10}$$

Calculate z from z=3\left(-\frac{9}{10}\right)+5\times \frac{9}{10}.

$$z=\frac{9}{5}$$

The system is now solved.

$$x=-\frac{9}{10} y=\frac{9}{10} z=\frac{9}{5}$$