Solve {c}{5x+3y+z=4}{x-y=z}{x+y=3}

$$\left. \begin{array} { c } { 5 x + 3 y + z = 4 } \\ { x - y = z } \\ { x + y = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{1}{2}=-0.5$$

$$y = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

$$z=-4$$

Short Steps Using Substitution

Reorder the equations.

$$x-y=z 5x+3y+z=4 x+y=3$$

Solve x-y=z for z.

$$z=x-y$$

Substitute x-y for z in the equation 5x+3y+z=4.

$$5x+3y+x-y=4$$

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

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

Substitute 2-3x for y in the equation x=-y+3.

$$x=-\left(2-3x\right)+3$$

Solve x=-\left(2-3x\right)+3 for x.

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

Substitute -\frac{1}{2} for x in the equation y=2-3x.

$$y=2-3\left(-\frac{1}{2}\right)$$

Calculate y from y=2-3\left(-\frac{1}{2}\right).

$$y=\frac{7}{2}$$

Substitute \frac{7}{2} for y and -\frac{1}{2} for x in the equation z=x-y.

$$z=-\frac{1}{2}-\frac{7}{2}$$

Calculate z from z=-\frac{1}{2}-\frac{7}{2}.

$$z=-4$$

The system is now solved.

$$x=-\frac{1}{2} y=\frac{7}{2} z=-4$$