Solve {l}{2.7x+13.6y+0z=408.24}{.6x+13.6y+22.68z=362.88}{.07x+13.6y+22.68z=317.52}

$$\left. \begin{array} { l } { 2.7 x + 13.6 y + 0 z = 408.24 } \\ { .6 x + 13.6 y + 22.68 z = 362.88 } \\ { .07 x + 13.6 y + 22.68 z = 317.52 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{4536}{53} = 85\frac{31}{53} \approx 85.58490566$$

$$y = \frac{117369}{9010} = 13\frac{239}{9010} \approx 13.026526082$$

$$z = \frac{314}{53} = 5\frac{49}{53} \approx 5.924528302$$

Short Steps Using Substitution

Solve 2.7x+13.6y+0z=408.24 for x.

$$x=-\frac{136}{27}y+151.2$$

Substitute -\frac{136}{27}y+151.2 for x in the second and third equation.

$$0.6\left(-\frac{136}{27}y+151.2\right)+13.6y+22.68z=362.88 0.07\left(-\frac{136}{27}y+151.2\right)+13.6y+22.68z=317.52$$

Solve these equations for y and z respectively.

$$y=\frac{2187}{85}-\frac{729}{340}z z=-\frac{8942}{15309}y+\frac{203}{15}$$

Substitute \frac{2187}{85}-\frac{729}{340}z for y in the equation z=-\frac{8942}{15309}y+\frac{203}{15}.

$$z=-\frac{8942}{15309}\left(\frac{2187}{85}-\frac{729}{340}z\right)+\frac{203}{15}$$

Solve z=-\frac{8942}{15309}\left(\frac{2187}{85}-\frac{729}{340}z\right)+\frac{203}{15} for z.

$$z=\frac{314}{53}$$

Substitute \frac{314}{53} for z in the equation y=\frac{2187}{85}-\frac{729}{340}z.

$$y=\frac{2187}{85}-\frac{729}{340}\times \frac{314}{53}$$

Calculate y from y=\frac{2187}{85}-\frac{729}{340}\times \frac{314}{53}.

$$y=\frac{117369}{9010}$$

Substitute \frac{117369}{9010} for y in the equation x=-\frac{136}{27}y+151.2.

$$x=-\frac{136}{27}\times \frac{117369}{9010}+151.2$$

Calculate x from x=-\frac{136}{27}\times \frac{117369}{9010}+151.2.

$$x=\frac{4536}{53}$$

The system is now solved.

$$x=\frac{4536}{53} y=\frac{117369}{9010} z=\frac{314}{53}$$