Solve {l}{1+0.54x+0.67y+0.47z=2.50}{.54+x+0.47y+0.43z=2.7}{0.17+0.47x+4+0.41z=2.5}

$$\left. \begin{array} { l } { 1 + 0.54 x + 0.67 y + 0.47 z = 2.50 } \\ { .54 + x + 0.47 y + 0.43 z = 2.7 } \\ { 0.17 + 0.47 x + 4 + 0.41 z = 2.5 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{208263}{69529} = 2\frac{69205}{69529} \approx 2.995340074$$

$$y = \frac{353948}{69529} = 5\frac{6303}{69529} \approx 5.090652821$$

$$z = -\frac{521944}{69529} = -7\frac{35241}{69529} \approx -7.506853255$$

Short Steps Using Substitution

Reorder the equations.

$$0.54+x+0.47y+0.43z=2.7 1+0.54x+0.67y+0.47z=2.5 0.17+0.47x+4+0.41z=2.5$$

Solve 0.54+x+0.47y+0.43z=2.7 for x.

$$x=2.16-0.47y-0.43z$$

Substitute 2.16-0.47y-0.43z for x in the second and third equation.

$$1+0.54\left(2.16-0.47y-0.43z\right)+0.67y+0.47z=2.5 0.17+0.47\left(2.16-0.47y-0.43z\right)+4+0.41z=2.5$$

Solve these equations for y and z respectively.

$$y=\frac{1668}{2081}-\frac{1189}{2081}z z=-\frac{3836}{297}+\frac{2209}{2079}y$$

Substitute \frac{1668}{2081}-\frac{1189}{2081}z for y in the equation z=-\frac{3836}{297}+\frac{2209}{2079}y.

$$z=-\frac{3836}{297}+\frac{2209}{2079}\left(\frac{1668}{2081}-\frac{1189}{2081}z\right)$$

Solve z=-\frac{3836}{297}+\frac{2209}{2079}\left(\frac{1668}{2081}-\frac{1189}{2081}z\right) for z.

$$z=-\frac{521944}{69529}$$

Substitute -\frac{521944}{69529} for z in the equation y=\frac{1668}{2081}-\frac{1189}{2081}z.

$$y=\frac{1668}{2081}-\frac{1189}{2081}\left(-\frac{521944}{69529}\right)$$

Calculate y from y=\frac{1668}{2081}-\frac{1189}{2081}\left(-\frac{521944}{69529}\right).

$$y=\frac{353948}{69529}$$

Substitute \frac{353948}{69529} for y and -\frac{521944}{69529} for z in the equation x=2.16-0.47y-0.43z.

$$x=2.16-0.47\times \frac{353948}{69529}-0.43\left(-\frac{521944}{69529}\right)$$

Calculate x from x=2.16-0.47\times \frac{353948}{69529}-0.43\left(-\frac{521944}{69529}\right).

$$x=\frac{208263}{69529}$$

The system is now solved.

$$x=\frac{208263}{69529} y=\frac{353948}{69529} z=-\frac{521944}{69529}$$