Solve {l}{x+y+z=1}{x+2y+3z=5}{2x-y-z=3}

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

Solve for x, y, z

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

$$y = -\frac{14}{3} = -4\frac{2}{3} \approx -4.666666667$$

$$z = \frac{13}{3} = 4\frac{1}{3} \approx 4.333333333$$

Short Steps Using Substitution

Solve x+y+z=1 for x.

$$x=-y-z+1$$

Substitute -y-z+1 for x in the second and third equation.

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

Solve these equations for y and z respectively.

$$y=4-2z z=-y-\frac{1}{3}$$

Substitute 4-2z for y in the equation z=-y-\frac{1}{3}.

$$z=-\left(4-2z\right)-\frac{1}{3}$$

Solve z=-\left(4-2z\right)-\frac{1}{3} for z.

$$z=\frac{13}{3}$$

Substitute \frac{13}{3} for z in the equation y=4-2z.

$$y=4-2\times \frac{13}{3}$$

Calculate y from y=4-2\times \frac{13}{3}.

$$y=-\frac{14}{3}$$

Substitute -\frac{14}{3} for y and \frac{13}{3} for z in the equation x=-y-z+1.

$$x=-\left(-\frac{14}{3}\right)-\frac{13}{3}+1$$

Calculate x from x=-\left(-\frac{14}{3}\right)-\frac{13}{3}+1.

$$x=\frac{4}{3}$$

The system is now solved.

$$x=\frac{4}{3} y=-\frac{14}{3} z=\frac{13}{3}$$