Solve {l}{6x+3y+2z=12}{4x-y+4z=37}{10x+5y+3z=21}

$$\left\{ \begin{array} { l } { 6 x + 3 y + 2 z = 12 } \\ { 4 x - y + 4 z = 37 } \\ { 10 x + 5 y + 3 z = 21 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{55}{6} = 9\frac{1}{6} \approx 9.166666667$$

$$y = -\frac{37}{3} = -12\frac{1}{3} \approx -12.333333333$$

$$z=-3$$

Short Steps Using Substitution

Reorder the equations.

$$4x-y+4z=37 6x+3y+2z=12 10x+5y+3z=21$$

Solve 4x-y+4z=37 for y.

$$y=4x+4z-37$$

Substitute 4x+4z-37 for y in the second and third equation.

$$6x+3\left(4x+4z-37\right)+2z=12 10x+5\left(4x+4z-37\right)+3z=21$$

Solve these equations for x and z respectively.

$$x=-\frac{7}{9}z+\frac{41}{6} z=\frac{206}{23}-\frac{30}{23}x$$

Substitute -\frac{7}{9}z+\frac{41}{6} for x in the equation z=\frac{206}{23}-\frac{30}{23}x.

$$z=\frac{206}{23}-\frac{30}{23}\left(-\frac{7}{9}z+\frac{41}{6}\right)$$

Solve z=\frac{206}{23}-\frac{30}{23}\left(-\frac{7}{9}z+\frac{41}{6}\right) for z.

$$z=-3$$

Substitute -3 for z in the equation x=-\frac{7}{9}z+\frac{41}{6}.

$$x=-\frac{7}{9}\left(-3\right)+\frac{41}{6}$$

Calculate x from x=-\frac{7}{9}\left(-3\right)+\frac{41}{6}.

$$x=\frac{55}{6}$$

Substitute \frac{55}{6} for x and -3 for z in the equation y=4x+4z-37.

$$y=4\times \frac{55}{6}+4\left(-3\right)-37$$

Calculate y from y=4\times \frac{55}{6}+4\left(-3\right)-37.

$$y=-\frac{37}{3}$$

The system is now solved.

$$x=\frac{55}{6} y=-\frac{37}{3} z=-3$$