Solve {l}{6x-2y+3z=17}{5x-3y+2z=11}{2x-9y-4z=-16}

$$\left\{ \begin{array} { l } { 6 x - 2 y + 3 z = 17 } \\ { 5 x - 3 y + 2 z = 11 } \\ { 2 x - 9 y - 4 z = - 16 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=2$$

$$z=1$$

Short Steps Using Substitution

Solve 6x-2y+3z=17 for x.

$$x=\frac{1}{3}y-\frac{1}{2}z+\frac{17}{6}$$

Substitute \frac{1}{3}y-\frac{1}{2}z+\frac{17}{6} for x in the second and third equation.

$$5\left(\frac{1}{3}y-\frac{1}{2}z+\frac{17}{6}\right)-3y+2z=11 2\left(\frac{1}{3}y-\frac{1}{2}z+\frac{17}{6}\right)-9y-4z=-16$$

Solve these equations for y and z respectively.

$$y=\frac{19}{8}-\frac{3}{8}z z=\frac{13}{3}-\frac{5}{3}y$$

Substitute \frac{19}{8}-\frac{3}{8}z for y in the equation z=\frac{13}{3}-\frac{5}{3}y.

$$z=\frac{13}{3}-\frac{5}{3}\left(\frac{19}{8}-\frac{3}{8}z\right)$$

Solve z=\frac{13}{3}-\frac{5}{3}\left(\frac{19}{8}-\frac{3}{8}z\right) for z.

$$z=1$$

Substitute 1 for z in the equation y=\frac{19}{8}-\frac{3}{8}z.

$$y=\frac{19}{8}-\frac{3}{8}$$

Calculate y from y=\frac{19}{8}-\frac{3}{8}.

$$y=2$$

Substitute 2 for y and 1 for z in the equation x=\frac{1}{3}y-\frac{1}{2}z+\frac{17}{6}.

$$x=\frac{1}{3}\times 2-\frac{1}{2}+\frac{17}{6}$$

Calculate x from x=\frac{1}{3}\times 2-\frac{1}{2}+\frac{17}{6}.

$$x=3$$

The system is now solved.

$$x=3 y=2 z=1$$