Solve {l}{5x+3y+x=0}{z=4x+7y+10}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { 5 x + 3 y + x = 0 }\\ { z = 4 x + 7 y + 10 }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{a}{10}+1$$

$$y=\frac{a}{5}-2$$

$$z=a$$

Short Steps Using Substitution

Reorder the equations.

$$z=4x+7y+10 5x+3y+x=0 a=z$$

Substitute 4x+7y+10 for z in the equation a=z.

$$a=4x+7y+10$$

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

$$y=-2x x=-\frac{5}{2}-\frac{7}{4}y+\frac{1}{4}a$$

Substitute -2x for y in the equation x=-\frac{5}{2}-\frac{7}{4}y+\frac{1}{4}a.

$$x=-\frac{5}{2}-\frac{7}{4}\left(-2\right)x+\frac{1}{4}a$$

Solve x=-\frac{5}{2}-\frac{7}{4}\left(-2\right)x+\frac{1}{4}a for x.

$$x=1-\frac{1}{10}a$$

Substitute 1-\frac{1}{10}a for x in the equation y=-2x.

$$y=-2\left(1-\frac{1}{10}a\right)$$

Calculate y from y=-2\left(1-\frac{1}{10}a\right).

$$y=-2+\frac{1}{5}a$$

Substitute -2+\frac{1}{5}a for y and 1-\frac{1}{10}a for x in the equation z=4x+7y+10.

$$z=4\left(1-\frac{1}{10}a\right)+7\left(-2+\frac{1}{5}a\right)+10$$

Calculate z from z=4\left(1-\frac{1}{10}a\right)+7\left(-2+\frac{1}{5}a\right)+10.

$$z=a$$

The system is now solved.

$$x=1-\frac{1}{10}a y=-2+\frac{1}{5}a z=a$$