Solve {c}{3x-5y=2a}{2x+7y=a-18}{x+y=0}

$$\left. \begin{array} { c } { 3 x - 5 y = 2 a } \\ { 2 x + 7 y = a - 18 } \\ { x + y = 0 } \end{array} \right.$$

Solve for x, y, a

$$x=2$$

$$y=-2$$

$$a=8$$

Short Steps Using Substitution

Reorder the equations.

$$2x+7y=a-18 3x-5y=2a x+y=0$$

Solve 2x+7y=a-18 for a.

$$a=2x+7y+18$$

Substitute 2x+7y+18 for a in the equation 3x-5y=2a.

$$3x-5y=2\left(2x+7y+18\right)$$

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

$$y=-\frac{1}{19}x-\frac{36}{19} x=-y$$

Substitute -\frac{1}{19}x-\frac{36}{19} for y in the equation x=-y.

$$x=-\left(-\frac{1}{19}x-\frac{36}{19}\right)$$

Solve x=-\left(-\frac{1}{19}x-\frac{36}{19}\right) for x.

$$x=2$$

Substitute 2 for x in the equation y=-\frac{1}{19}x-\frac{36}{19}.

$$y=-\frac{1}{19}\times 2-\frac{36}{19}$$

Calculate y from y=-\frac{1}{19}\times 2-\frac{36}{19}.

$$y=-2$$

Substitute -2 for y and 2 for x in the equation a=2x+7y+18.

$$a=2\times 2+7\left(-2\right)+18$$

Calculate a from a=2\times 2+7\left(-2\right)+18.

$$a=8$$

The system is now solved.

$$x=2 y=-2 a=8$$