Solve {r}{x+y+z=1000}{y+2z=950}{2x+y+2z=1575}

$$\left\{ \begin{array} { r } { x + y + z = 1000 } \\ { y + 2 z = 950 } \\ { 2 x + y + 2 z = 1575 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{625}{2} = 312\frac{1}{2} = 312.5$$

$$y=425$$

$$z = \frac{525}{2} = 262\frac{1}{2} = 262.5$$

Short Steps Using Substitution

Solve x+y+z=1000 for x.

$$x=-y-z+1000$$

Substitute -y-z+1000 for x in the equation 2x+y+2z=1575.

$$2\left(-y-z+1000\right)+y+2z=1575$$

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

$$z=-\frac{1}{2}y+475 y=425$$

Substitute 425 for y in the equation z=-\frac{1}{2}y+475.

$$z=-\frac{1}{2}\times 425+475$$

Calculate z from z=-\frac{1}{2}\times 425+475.

$$z=\frac{525}{2}$$

Substitute \frac{525}{2} for z and 425 for y in the equation x=-y-z+1000.

$$x=-425-\frac{525}{2}+1000$$

Calculate x from x=-425-\frac{525}{2}+1000.

$$x=\frac{625}{2}$$

The system is now solved.

$$x=\frac{625}{2} y=425 z=\frac{525}{2}$$