Solve {r}{5z-y=2x-3}{7x+z=2+5y}{x+3y+2z=2(1-x)}

$$\left.\begin{array} { r } { 5 z - y = 2 x - 3 } \\ { 7 x + z = 2 + 5 y } \\ { x + 3 y + 2 z = 2 ( 1 - x ) } \end{array} \right\}$$

Solve for z, y, x

$$x=\frac{121}{217}\approx 0.557603687$$

$$y=\frac{69}{217}\approx 0.31797235$$

$$z=-\frac{68}{217}\approx -0.313364055$$

Short Steps Using Substitution

Solve 5z-y=2x-3 for y.

$$y=5z-2x+3$$

Substitute 5z-2x+3 for y in the second and third equation.

$$7x+z=2+5\left(5z-2x+3\right) x+3\left(5z-2x+3\right)+2z=2\left(1-x\right)$$

Solve these equations for z and x respectively.

$$z=-\frac{17}{24}+\frac{17}{24}x x=\frac{17}{3}z+\frac{7}{3}$$

Substitute -\frac{17}{24}+\frac{17}{24}x for z in the equation x=\frac{17}{3}z+\frac{7}{3}.

$$x=\frac{17}{3}\left(-\frac{17}{24}+\frac{17}{24}x\right)+\frac{7}{3}$$

Solve x=\frac{17}{3}\left(-\frac{17}{24}+\frac{17}{24}x\right)+\frac{7}{3} for x.

$$x=\frac{121}{217}$$

Substitute \frac{121}{217} for x in the equation z=-\frac{17}{24}+\frac{17}{24}x.

$$z=-\frac{17}{24}+\frac{17}{24}\times \frac{121}{217}$$

Calculate z from z=-\frac{17}{24}+\frac{17}{24}\times \frac{121}{217}.

$$z=-\frac{68}{217}$$

Substitute -\frac{68}{217} for z and \frac{121}{217} for x in the equation y=5z-2x+3.

$$y=5\left(-\frac{68}{217}\right)-2\times \frac{121}{217}+3$$

Calculate y from y=5\left(-\frac{68}{217}\right)-2\times \frac{121}{217}+3.

$$y=\frac{69}{217}$$

The system is now solved.

$$z=-\frac{68}{217} y=\frac{69}{217} x=\frac{121}{217}$$