Solve {l}{x-y+3z=11}{4x+y-z=-4}{2x+y+3z=10}

$$\left. \begin{array} { l } { x - y + 3 z = 11 } \\ { 4 x + y - z = - 4 } \\ { 2 x + y + 3 z = 10 } \end{array} \right.$$

Solve for x, y, z

$$x=0$$

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

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

Short Steps Using Substitution

Solve x-y+3z=11 for x.

$$x=y-3z+11$$

Substitute y-3z+11 for x in the second and third equation.

$$4\left(y-3z+11\right)+y-z=-4 2\left(y-3z+11\right)+y+3z=10$$

Solve these equations for y and z respectively.

$$y=\frac{13}{5}z-\frac{48}{5} z=4+y$$

Substitute \frac{13}{5}z-\frac{48}{5} for y in the equation z=4+y.

$$z=4+\frac{13}{5}z-\frac{48}{5}$$

Solve z=4+\frac{13}{5}z-\frac{48}{5} for z.

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

Substitute \frac{7}{2} for z in the equation y=\frac{13}{5}z-\frac{48}{5}.

$$y=\frac{13}{5}\times \frac{7}{2}-\frac{48}{5}$$

Calculate y from y=\frac{13}{5}\times \frac{7}{2}-\frac{48}{5}.

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

Substitute -\frac{1}{2} for y and \frac{7}{2} for z in the equation x=y-3z+11.

$$x=-\frac{1}{2}-3\times \frac{7}{2}+11$$

Calculate x from x=-\frac{1}{2}-3\times \frac{7}{2}+11.

$$x=0$$

The system is now solved.

$$x=0 y=-\frac{1}{2} z=\frac{7}{2}$$