Solve {l}{2x+2y+1z=600}{2x+y+2z=660}{3x+2y+2z=870}

$$\left. \begin{array} { l } { 2 x + 2 y + 1 z = 600 } \\ { 2 x + y + 2 z = 660 } \\ { 3 x + 2 y + 2 z = 870 } \end{array} \right.$$

Solve for x, y, z

$$x=90$$

$$y=120$$

$$z=180$$

Short Steps Using Substitution

Reorder the equations.

$$2x+y+2z=660 2x+2y+1z=600 3x+2y+2z=870$$

Solve 2x+y+2z=660 for y.

$$y=660-2x-2z$$

Substitute 660-2x-2z for y in the second and third equation.

$$2x+2\left(660-2x-2z\right)+1z=600 3x+2\left(660-2x-2z\right)+2z=870$$

Solve these equations for x and z respectively.

$$x=360-\frac{3}{2}z z=-\frac{1}{2}x+225$$

Substitute 360-\frac{3}{2}z for x in the equation z=-\frac{1}{2}x+225.

$$z=-\frac{1}{2}\left(360-\frac{3}{2}z\right)+225$$

Solve z=-\frac{1}{2}\left(360-\frac{3}{2}z\right)+225 for z.

$$z=180$$

Substitute 180 for z in the equation x=360-\frac{3}{2}z.

$$x=360-\frac{3}{2}\times 180$$

Calculate x from x=360-\frac{3}{2}\times 180.

$$x=90$$

Substitute 90 for x and 180 for z in the equation y=660-2x-2z.

$$y=660-2\times 90-2\times 180$$

Calculate y from y=660-2\times 90-2\times 180.

$$y=120$$

The system is now solved.

$$x=90 y=120 z=180$$