Solve {c}{2x+4y+6z=18}{4x+5y-6z=24}{3x+y-2z=4}

$$\left. \begin{array} { c } { 2 x + 4 y + 6 z = 18 } \\ { 4 x + 5 y - 6 z = 24 } \\ { 3 x + y - 2 z = 4 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{8}{19}\approx -0.421052632$$

$$y = \frac{94}{19} = 4\frac{18}{19} \approx 4.947368421$$

$$z=-\frac{3}{19}\approx -0.157894737$$

Short Steps Using Substitution

Reorder the equations.

$$3x+y-2z=4 4x+5y-6z=24 2x+4y+6z=18$$

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

$$y=-3x+2z+4$$

Substitute -3x+2z+4 for y in the second and third equation.

$$4x+5\left(-3x+2z+4\right)-6z=24 2x+4\left(-3x+2z+4\right)+6z=18$$

Solve these equations for x and z respectively.

$$x=-\frac{4}{11}+\frac{4}{11}z z=\frac{1}{7}+\frac{5}{7}x$$

Substitute -\frac{4}{11}+\frac{4}{11}z for x in the equation z=\frac{1}{7}+\frac{5}{7}x.

$$z=\frac{1}{7}+\frac{5}{7}\left(-\frac{4}{11}+\frac{4}{11}z\right)$$

Solve z=\frac{1}{7}+\frac{5}{7}\left(-\frac{4}{11}+\frac{4}{11}z\right) for z.

$$z=-\frac{3}{19}$$

Substitute -\frac{3}{19} for z in the equation x=-\frac{4}{11}+\frac{4}{11}z.

$$x=-\frac{4}{11}+\frac{4}{11}\left(-\frac{3}{19}\right)$$

Calculate x from x=-\frac{4}{11}+\frac{4}{11}\left(-\frac{3}{19}\right).

$$x=-\frac{8}{19}$$

Substitute -\frac{8}{19} for x and -\frac{3}{19} for z in the equation y=-3x+2z+4.

$$y=-3\left(-\frac{8}{19}\right)+2\left(-\frac{3}{19}\right)+4$$

Calculate y from y=-3\left(-\frac{8}{19}\right)+2\left(-\frac{3}{19}\right)+4.

$$y=\frac{94}{19}$$

The system is now solved.

$$x=-\frac{8}{19} y=\frac{94}{19} z=-\frac{3}{19}$$