Solve {r}{x+y-z=0}{2x+5y+3z=210}{3x-y+2z=70}

$$\left\{ \begin{array} { r } { x + y - z = 0 } \\ { 2 x + 5 y + 3 z = 210 } \\ { 3 x - y + 2 z = 70 } \end{array} \right.$$

Solve for x, y, z

$$x=10$$

$$y=20$$

$$z=30$$

Short Steps Using Substitution

Solve x+y-z=0 for x.

$$x=-y+z$$

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

$$2\left(-y+z\right)+5y+3z=210 3\left(-y+z\right)-y+2z=70$$

Solve these equations for y and z respectively.

$$y=70-\frac{5}{3}z z=14+\frac{4}{5}y$$

Substitute 70-\frac{5}{3}z for y in the equation z=14+\frac{4}{5}y.

$$z=14+\frac{4}{5}\left(70-\frac{5}{3}z\right)$$

Solve z=14+\frac{4}{5}\left(70-\frac{5}{3}z\right) for z.

$$z=30$$

Substitute 30 for z in the equation y=70-\frac{5}{3}z.

$$y=70-\frac{5}{3}\times 30$$

Calculate y from y=70-\frac{5}{3}\times 30.

$$y=20$$

Substitute 20 for y and 30 for z in the equation x=-y+z.

$$x=-20+30$$

Calculate x from x=-20+30.

$$x=10$$

The system is now solved.

$$x=10 y=20 z=30$$