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

$$x = -\frac{15}{4} = -3\frac{3}{4} = -3.75$$

$$y=\frac{3}{4}=0.75$$

$$r = \frac{43}{4} = 10\frac{3}{4} = 10.75$$

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.

$$y=-\frac{1}{13}+\frac{1}{13}r r=\frac{79}{7}-\frac{5}{7}y$$

Substitute -\frac{1}{13}+\frac{1}{13}r for y in the equation r=\frac{79}{7}-\frac{5}{7}y.

$$r=\frac{79}{7}-\frac{5}{7}\left(-\frac{1}{13}+\frac{1}{13}r\right)$$

Solve r=\frac{79}{7}-\frac{5}{7}\left(-\frac{1}{13}+\frac{1}{13}r\right) for r.

$$r=\frac{43}{4}$$

Substitute \frac{43}{4} for r in the equation y=-\frac{1}{13}+\frac{1}{13}r.

$$y=-\frac{1}{13}+\frac{1}{13}\times \frac{43}{4}$$

Calculate y from y=-\frac{1}{13}+\frac{1}{13}\times \frac{43}{4}.

$$y=\frac{3}{4}$$

Substitute \frac{3}{4} for y and \frac{43}{4} for r in the equation x=\frac{1}{2}+\frac{3}{2}y-\frac{1}{2}r.

$$x=\frac{1}{2}+\frac{3}{2}\times \frac{3}{4}-\frac{1}{2}\times \frac{43}{4}$$

Calculate x from x=\frac{1}{2}+\frac{3}{2}\times \frac{3}{4}-\frac{1}{2}\times \frac{43}{4}.

$$x=-\frac{15}{4}$$

The system is now solved.

$$x=-\frac{15}{4} y=\frac{3}{4} r=\frac{43}{4}$$