Solve {l}{5x-y-15=0}{z=x-y}{2y-x-5=0}

$$\left\{ \begin{array} { l } { 5 x - y - 15 = 0 } \\ { z = x - y } \\ { 2 y - x - 5 = 0 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{35}{9} = 3\frac{8}{9} \approx 3.888888889$$

$$y = \frac{40}{9} = 4\frac{4}{9} \approx 4.444444444$$

$$z=-\frac{5}{9}\approx -0.555555556$$

Short Steps Using Substitution

Reorder the equations.

$$z=x-y 5x-y-15=0 2y-x-5=0$$

Solve the second equation for y and the third equation for x.

$$y=-15+5x x=2y-5$$

Substitute -15+5x for y in the equation x=2y-5.

$$x=2\left(-15+5x\right)-5$$

Solve x=2\left(-15+5x\right)-5 for x.

$$x=\frac{35}{9}$$

Substitute \frac{35}{9} for x in the equation y=-15+5x.

$$y=-15+5\times \frac{35}{9}$$

Calculate y from y=-15+5\times \frac{35}{9}.

$$y=\frac{40}{9}$$

Substitute \frac{40}{9} for y and \frac{35}{9} for x in the equation z=x-y.

$$z=\frac{35}{9}-\frac{40}{9}$$

Calculate z from z=\frac{35}{9}-\frac{40}{9}.

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

The system is now solved.

$$x=\frac{35}{9} y=\frac{40}{9} z=-\frac{5}{9}$$