Solve {l}{x+y+z=673}{1/2(x+z)+2f=y}{x-8=z(3)}

$$\left\{ \begin{array} { l } { x + y + z = 673 } \\ { \frac { 1 } { 2 } ( x + z ) + 2 f = y } \\ { x - 8 = z ( 3 ) } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{677}{2}-f$$

$$y=\frac{4f+673}{3}$$

$$z=-\frac{f}{3}+\frac{661}{6}$$

Short Steps Using Substitution

Reorder the equations.

$$\frac{1}{2}\left(x+z\right)+2f=y x+y+z=673 x-8=z\times 3$$

Solve \frac{1}{2}\left(x+z\right)+2f=y for y.

$$y=\frac{1}{2}x+\frac{1}{2}z+2f$$

Substitute \frac{1}{2}x+\frac{1}{2}z+2f for y in the equation x+y+z=673.

$$x+\frac{1}{2}x+\frac{1}{2}z+2f+z=673$$

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

$$x=-z-\frac{4}{3}f+\frac{1346}{3} z=-\frac{8}{3}+\frac{1}{3}x$$

Substitute -z-\frac{4}{3}f+\frac{1346}{3} for x in the equation z=-\frac{8}{3}+\frac{1}{3}x.

$$z=-\frac{8}{3}+\frac{1}{3}\left(-z-\frac{4}{3}f+\frac{1346}{3}\right)$$

Solve z=-\frac{8}{3}+\frac{1}{3}\left(-z-\frac{4}{3}f+\frac{1346}{3}\right) for z.

$$z=\frac{661}{6}-\frac{1}{3}f$$

Substitute \frac{661}{6}-\frac{1}{3}f for z in the equation x=-z-\frac{4}{3}f+\frac{1346}{3}.

$$x=-\left(\frac{661}{6}-\frac{1}{3}f\right)-\frac{4}{3}f+\frac{1346}{3}$$

Calculate x from x=-\left(\frac{661}{6}-\frac{1}{3}f\right)-\frac{4}{3}f+\frac{1346}{3}.

$$x=\frac{677}{2}-f$$

Substitute \frac{677}{2}-f for x and \frac{661}{6}-\frac{1}{3}f for z in the equation y=\frac{1}{2}x+\frac{1}{2}z+2f.

$$y=\frac{1}{2}\left(\frac{677}{2}-f\right)+\frac{1}{2}\left(\frac{661}{6}-\frac{1}{3}f\right)+2f$$

Calculate y from y=\frac{1}{2}\left(\frac{677}{2}-f\right)+\frac{1}{2}\left(\frac{661}{6}-\frac{1}{3}f\right)+2f.

$$y=\frac{673}{3}+\frac{4}{3}f$$

The system is now solved.

$$x=\frac{677}{2}-f y=\frac{673}{3}+\frac{4}{3}f z=\frac{661}{6}-\frac{1}{3}f$$