Solve {l}{x+y=z}{y-x=N}{z=5}

$$\left. \begin{array} { l } { x + y = z } \\ { y - x = N } \\ { z = 5 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{5-N}{2}$$

$$y=\frac{N+5}{2}$$

$$z=5$$

Short Steps Using Substitution

Reorder the equations.

$$z=5 y-x=N x+y=z$$

Substitute 5 for z in the equation x+y=z.

$$x+y=5$$

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

$$y=x+N x=-y+5$$

Substitute x+N for y in the equation x=-y+5.

$$x=-\left(x+N\right)+5$$

Solve x=-\left(x+N\right)+5 for x.

$$x=\frac{5}{2}-\frac{1}{2}N$$

Substitute \frac{5}{2}-\frac{1}{2}N for x in the equation y=x+N.

$$y=\frac{5}{2}-\frac{1}{2}N+N$$

Calculate y from y=\frac{5}{2}-\frac{1}{2}N+N.

$$y=\frac{5}{2}+\frac{1}{2}N$$

The system is now solved.

$$x=\frac{5}{2}-\frac{1}{2}N y=\frac{5}{2}+\frac{1}{2}N z=5$$