Solve {l}{x+y+2z=4}{2x-3y+z=0}{5x-y-z=3}

$$\left\{ \begin{array} { l } { x + y + 2 z = 4 } \\ { 2 x - 3 y + z = 0 } \\ { 5 x - y - z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=1$$

$$z=1$$

Short Steps Using Substitution

Solve x+y+2z=4 for x.

$$x=-y-2z+4$$

Substitute -y-2z+4 for x in the second and third equation.

$$2\left(-y-2z+4\right)-3y+z=0 5\left(-y-2z+4\right)-y-z=3$$

Solve these equations for y and z respectively.

$$y=-\frac{3}{5}z+\frac{8}{5} z=-\frac{6}{11}y+\frac{17}{11}$$

Substitute -\frac{3}{5}z+\frac{8}{5} for y in the equation z=-\frac{6}{11}y+\frac{17}{11}.

$$z=-\frac{6}{11}\left(-\frac{3}{5}z+\frac{8}{5}\right)+\frac{17}{11}$$

Solve z=-\frac{6}{11}\left(-\frac{3}{5}z+\frac{8}{5}\right)+\frac{17}{11} for z.

$$z=1$$

Substitute 1 for z in the equation y=-\frac{3}{5}z+\frac{8}{5}.

$$y=-\frac{3}{5}+\frac{8}{5}$$

Calculate y from y=-\frac{3}{5}+\frac{8}{5}.

$$y=1$$

Substitute 1 for y and 1 for z in the equation x=-y-2z+4.

$$x=-1-2+4$$

Calculate x from x=-1-2+4.

$$x=1$$

The system is now solved.

$$x=1 y=1 z=1$$