Solve {l}{5x+4y+6z=500}{4x+2y+2z=250}{3x+3y+2z=325}

$$\left. \begin{array} { l } { 5 x + 4 y + 6 z = 500 } \\ { 4 x + 2 y + 2 z = 250 } \\ { 3 x + 3 y + 2 z = 325 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{100}{9} = 11\frac{1}{9} \approx 11.111111111$$

$$y = \frac{775}{9} = 86\frac{1}{9} \approx 86.111111111$$

$$z = \frac{50}{3} = 16\frac{2}{3} \approx 16.666666667$$

Short Steps Using Substitution

Solve 5x+4y+6z=500 for x.

$$x=100-\frac{4}{5}y-\frac{6}{5}z$$

Substitute 100-\frac{4}{5}y-\frac{6}{5}z for x in the second and third equation.

$$4\left(100-\frac{4}{5}y-\frac{6}{5}z\right)+2y+2z=250 3\left(100-\frac{4}{5}y-\frac{6}{5}z\right)+3y+2z=325$$

Solve these equations for y and z respectively.

$$y=125-\frac{7}{3}z z=-\frac{125}{8}+\frac{3}{8}y$$

Substitute 125-\frac{7}{3}z for y in the equation z=-\frac{125}{8}+\frac{3}{8}y.

$$z=-\frac{125}{8}+\frac{3}{8}\left(125-\frac{7}{3}z\right)$$

Solve z=-\frac{125}{8}+\frac{3}{8}\left(125-\frac{7}{3}z\right) for z.

$$z=\frac{50}{3}$$

Substitute \frac{50}{3} for z in the equation y=125-\frac{7}{3}z.

$$y=125-\frac{7}{3}\times \frac{50}{3}$$

Calculate y from y=125-\frac{7}{3}\times \frac{50}{3}.

$$y=\frac{775}{9}$$

Substitute \frac{775}{9} for y and \frac{50}{3} for z in the equation x=100-\frac{4}{5}y-\frac{6}{5}z.

$$x=100-\frac{4}{5}\times \frac{775}{9}-\frac{6}{5}\times \frac{50}{3}$$

Calculate x from x=100-\frac{4}{5}\times \frac{775}{9}-\frac{6}{5}\times \frac{50}{3}.

$$x=\frac{100}{9}$$

The system is now solved.

$$x=\frac{100}{9} y=\frac{775}{9} z=\frac{50}{3}$$