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

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

Solve for x, y, z

$$x = -\frac{251}{78} = -3\frac{17}{78} \approx -3.217948718$$

$$y = \frac{49}{39} = 1\frac{10}{39} \approx 1.256410256$$

$$z = -\frac{13}{6} = -2\frac{1}{6} \approx -2.166666667$$

Short Steps Using Substitution

Reorder the equations.

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

Solve -3x+2y+z=10 for z.

$$z=3x-2y+10$$

Substitute 3x-2y+10 for z in the second and third equation.

$$2x+3y-4\left(3x-2y+10\right)=6 -x+5y+3\left(3x-2y+10\right)=3$$

Solve these equations for y and x respectively.

$$y=\frac{46}{11}+\frac{10}{11}x x=-\frac{27}{8}+\frac{1}{8}y$$

Substitute \frac{46}{11}+\frac{10}{11}x for y in the equation x=-\frac{27}{8}+\frac{1}{8}y.

$$x=-\frac{27}{8}+\frac{1}{8}\left(\frac{46}{11}+\frac{10}{11}x\right)$$

Solve x=-\frac{27}{8}+\frac{1}{8}\left(\frac{46}{11}+\frac{10}{11}x\right) for x.

$$x=-\frac{251}{78}$$

Substitute -\frac{251}{78} for x in the equation y=\frac{46}{11}+\frac{10}{11}x.

$$y=\frac{46}{11}+\frac{10}{11}\left(-\frac{251}{78}\right)$$

Calculate y from y=\frac{46}{11}+\frac{10}{11}\left(-\frac{251}{78}\right).

$$y=\frac{49}{39}$$

Substitute \frac{49}{39} for y and -\frac{251}{78} for x in the equation z=3x-2y+10.

$$z=3\left(-\frac{251}{78}\right)-2\times \frac{49}{39}+10$$

Calculate z from z=3\left(-\frac{251}{78}\right)-2\times \frac{49}{39}+10.

$$z=-\frac{13}{6}$$

The system is now solved.

$$x=-\frac{251}{78} y=\frac{49}{39} z=-\frac{13}{6}$$