Solve {c}{x+y+z=15000}{9x+6y+5z=89000}{4.5x+6y+2.5z=50500}

$$\left. \begin{array} { c } { x + y + z = 15000 } \\ { 9 x + 6 y + 5 z = 89000 } \\ { 4.5 x + 6 y + 2.5 z = 50500 } \end{array} \right.$$

Solve for x, y, z

$$x=3000$$

$$y=2000$$

$$z=10000$$

Short Steps Using Substitution

Solve x+y+z=15000 for x.

$$x=-y-z+15000$$

Substitute -y-z+15000 for x in the second and third equation.

$$9\left(-y-z+15000\right)+6y+5z=89000 4.5\left(-y-z+15000\right)+6y+2.5z=50500$$

Solve these equations for y and z respectively.

$$y=\frac{46000}{3}-\frac{4}{3}z z=\frac{3}{4}y+8500$$

Substitute \frac{46000}{3}-\frac{4}{3}z for y in the equation z=\frac{3}{4}y+8500.

$$z=\frac{3}{4}\left(\frac{46000}{3}-\frac{4}{3}z\right)+8500$$

Solve z=\frac{3}{4}\left(\frac{46000}{3}-\frac{4}{3}z\right)+8500 for z.

$$z=10000$$

Substitute 10000 for z in the equation y=\frac{46000}{3}-\frac{4}{3}z.

$$y=\frac{46000}{3}-\frac{4}{3}\times 10000$$

Calculate y from y=\frac{46000}{3}-\frac{4}{3}\times 10000.

$$y=2000$$

Substitute 2000 for y and 10000 for z in the equation x=-y-z+15000.

$$x=-2000-10000+15000$$

Calculate x from x=-2000-10000+15000.

$$x=3000$$

The system is now solved.

$$x=3000 y=2000 z=10000$$