Solve {l}{x+y+z=3}{x-y-z=4}{x+2y+z=5}

$$\left. \begin{array} { l } { x + y + z = 3 } \\ { x - y - z = 4 } \\ { x + 2 y + z = 5 } \end{array} \right.$$

Solve for x, y, z

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

$$y=2$$

$$z = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

Short Steps Using Substitution

Solve x+y+z=3 for x.

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

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

$$-y-z+3-y-z=4 -y-z+3+2y+z=5$$

Solve these equations for z and y respectively.

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

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

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

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

$$z=-\frac{5}{2}$$

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

$$x=-2-\left(-\frac{5}{2}\right)+3$$

Calculate x from x=-2-\left(-\frac{5}{2}\right)+3.

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

The system is now solved.

$$x=\frac{7}{2} y=2 z=-\frac{5}{2}$$