Solve {l}{2x-3y+4z=51}{3x+4y-2x=0}{-4x+2y+3z=8}

$$\left. \begin{array} { l } { 2 x - 3 y + 4 z = 51 } \\ { 3 x + 4 y - 2 x = 0 } \\ { - 4 x + 2 y + 3 z = 8 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{484}{105} = 4\frac{64}{105} \approx 4.60952381$$

$$y = -\frac{121}{105} = -1\frac{16}{105} \approx -1.152380952$$

$$z = \frac{1006}{105} = 9\frac{61}{105} \approx 9.580952381$$

Short Steps Using Substitution

Solve 2x-3y+4z=51 for x.

$$x=\frac{3}{2}y-2z+\frac{51}{2}$$

Substitute \frac{3}{2}y-2z+\frac{51}{2} for x in the second and third equation.

$$3\left(\frac{3}{2}y-2z+\frac{51}{2}\right)+4y-2\left(\frac{3}{2}y-2z+\frac{51}{2}\right)=0 -4\left(\frac{3}{2}y-2z+\frac{51}{2}\right)+2y+3z=8$$

Solve these equations for y and z respectively.

$$y=-\frac{51}{11}+\frac{4}{11}z z=10+\frac{4}{11}y$$

Substitute -\frac{51}{11}+\frac{4}{11}z for y in the equation z=10+\frac{4}{11}y.

$$z=10+\frac{4}{11}\left(-\frac{51}{11}+\frac{4}{11}z\right)$$

Solve z=10+\frac{4}{11}\left(-\frac{51}{11}+\frac{4}{11}z\right) for z.

$$z=\frac{1006}{105}$$

Substitute \frac{1006}{105} for z in the equation y=-\frac{51}{11}+\frac{4}{11}z.

$$y=-\frac{51}{11}+\frac{4}{11}\times \frac{1006}{105}$$

Calculate y from y=-\frac{51}{11}+\frac{4}{11}\times \frac{1006}{105}.

$$y=-\frac{121}{105}$$

Substitute -\frac{121}{105} for y and \frac{1006}{105} for z in the equation x=\frac{3}{2}y-2z+\frac{51}{2}.

$$x=\frac{3}{2}\left(-\frac{121}{105}\right)-2\times \frac{1006}{105}+\frac{51}{2}$$

Calculate x from x=\frac{3}{2}\left(-\frac{121}{105}\right)-2\times \frac{1006}{105}+\frac{51}{2}.

$$x=\frac{484}{105}$$

The system is now solved.

$$x=\frac{484}{105} y=-\frac{121}{105} z=\frac{1006}{105}$$