Solve {l}{x-3y+2z=-3}{5x+6y-z=13}{4x-y+3z=8}

$$\left. \begin{array} { l } { x - 3 y + 2 z = - 3 } \\ { 5 x + 6 y - z = 13 } \\ { 4 x - y + 3 z = 8 } \end{array} \right.$$

Solve for x, y, z

$$x=-2$$

$$y=5$$

$$z=7$$

Short Steps Using Substitution

Solve x-3y+2z=-3 for x.

$$x=3y-2z-3$$

Substitute 3y-2z-3 for x in the second and third equation.

$$5\left(3y-2z-3\right)+6y-z=13 4\left(3y-2z-3\right)-y+3z=8$$

Solve these equations for y and z respectively.

$$y=\frac{11}{21}z+\frac{4}{3} z=\frac{11}{5}y-4$$

Substitute \frac{11}{21}z+\frac{4}{3} for y in the equation z=\frac{11}{5}y-4.

$$z=\frac{11}{5}\left(\frac{11}{21}z+\frac{4}{3}\right)-4$$

Solve z=\frac{11}{5}\left(\frac{11}{21}z+\frac{4}{3}\right)-4 for z.

$$z=7$$

Substitute 7 for z in the equation y=\frac{11}{21}z+\frac{4}{3}.

$$y=\frac{11}{21}\times 7+\frac{4}{3}$$

Calculate y from y=\frac{11}{21}\times 7+\frac{4}{3}.

$$y=5$$

Substitute 5 for y and 7 for z in the equation x=3y-2z-3.

$$x=3\times 5-2\times 7-3$$

Calculate x from x=3\times 5-2\times 7-3.

$$x=-2$$

The system is now solved.

$$x=-2 y=5 z=7$$