Solve {l}{10x+15y+5r=5}{15x+10y+5r=5}{5x+5y+20r=200}

$$\left. \begin{array} { l } { 10 x + 15 y + 5 r = 5 } \\ { 15 x + 10 y + 5 r = 5 } \\ { 5 x + 5 y + 20 r = 200 } \end{array} \right.$$

Solve for x, y, r

$$x=-2$$

$$y=-2$$

$$r=11$$

Short Steps Using Substitution

Solve 10x+15y+5r=5 for x.

$$x=\frac{1}{2}-\frac{3}{2}y-\frac{1}{2}r$$

Substitute \frac{1}{2}-\frac{3}{2}y-\frac{1}{2}r for x in the second and third equation.

$$15\left(\frac{1}{2}-\frac{3}{2}y-\frac{1}{2}r\right)+10y+5r=5 5\left(\frac{1}{2}-\frac{3}{2}y-\frac{1}{2}r\right)+5y+20r=200$$

Solve these equations for y and r respectively.

$$y=\frac{1}{5}-\frac{1}{5}r r=\frac{79}{7}+\frac{1}{7}y$$

Substitute \frac{1}{5}-\frac{1}{5}r for y in the equation r=\frac{79}{7}+\frac{1}{7}y.

$$r=\frac{79}{7}+\frac{1}{7}\left(\frac{1}{5}-\frac{1}{5}r\right)$$

Solve r=\frac{79}{7}+\frac{1}{7}\left(\frac{1}{5}-\frac{1}{5}r\right) for r.

$$r=11$$

Substitute 11 for r in the equation y=\frac{1}{5}-\frac{1}{5}r.

$$y=\frac{1}{5}-\frac{1}{5}\times 11$$

Calculate y from y=\frac{1}{5}-\frac{1}{5}\times 11.

$$y=-2$$

Substitute -2 for y and 11 for r in the equation x=\frac{1}{2}-\frac{3}{2}y-\frac{1}{2}r.

$$x=\frac{1}{2}-\frac{3}{2}\left(-2\right)-\frac{1}{2}\times 11$$

Calculate x from x=\frac{1}{2}-\frac{3}{2}\left(-2\right)-\frac{1}{2}\times 11.

$$x=-2$$

The system is now solved.

$$x=-2 y=-2 r=11$$