Solve {l}{2x+3y+4z=3}{x-2y+z=9}{5x+y-2=12}

$$\left. \begin{array} { l } { 2 x + 3 y + 4 z = 3 } \\ { x - 2 y + z = 9 } \\ { 5 x + y - 2 = 12 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{187}{57} = 3\frac{16}{57} \approx 3.280701754$$

$$y = -\frac{137}{57} = -2\frac{23}{57} \approx -2.403508772$$

$$z=\frac{52}{57}\approx 0.912280702$$

Short Steps Using Substitution

Reorder the equations.

$$x-2y+z=9 2x+3y+4z=3 5x+y-2=12$$

Solve x-2y+z=9 for x.

$$x=2y-z+9$$

Substitute 2y-z+9 for x in the second and third equation.

$$2\left(2y-z+9\right)+3y+4z=3 5\left(2y-z+9\right)+y-2=12$$

Solve these equations for y and z respectively.

$$y=-\frac{2}{7}z-\frac{15}{7} z=\frac{31}{5}+\frac{11}{5}y$$

Substitute -\frac{2}{7}z-\frac{15}{7} for y in the equation z=\frac{31}{5}+\frac{11}{5}y.

$$z=\frac{31}{5}+\frac{11}{5}\left(-\frac{2}{7}z-\frac{15}{7}\right)$$

Solve z=\frac{31}{5}+\frac{11}{5}\left(-\frac{2}{7}z-\frac{15}{7}\right) for z.

$$z=\frac{52}{57}$$

Substitute \frac{52}{57} for z in the equation y=-\frac{2}{7}z-\frac{15}{7}.

$$y=-\frac{2}{7}\times \frac{52}{57}-\frac{15}{7}$$

Calculate y from y=-\frac{2}{7}\times \frac{52}{57}-\frac{15}{7}.

$$y=-\frac{137}{57}$$

Substitute -\frac{137}{57} for y and \frac{52}{57} for z in the equation x=2y-z+9.

$$x=2\left(-\frac{137}{57}\right)-\frac{52}{57}+9$$

Calculate x from x=2\left(-\frac{137}{57}\right)-\frac{52}{57}+9.

$$x=\frac{187}{57}$$

The system is now solved.

$$x=\frac{187}{57} y=-\frac{137}{57} z=\frac{52}{57}$$