Solve {l}{22,7x+13,6y+0z=408,24}{13,6x+13,6y+22,68z=362,88}{9.07x+13,6y+22,68z=317,52}

$$\left. \begin{array} { l } { 22,7 x + 13,6 y + 0 z = 408,24 } \\ { 13,6 x + 13,6 y + 22,68 z = 362,88 } \\ { 9.07 x + 13,6 y + 22,68 z = 317,52 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{1512}{151} = 10\frac{2}{151} \approx 10.013245033$$

$$y = \frac{341523}{25670} = 13\frac{7813}{25670} \approx 13.30436307$$

$$z = \frac{914}{453} = 2\frac{8}{453} \approx 2.017660044$$

Short Steps Using Substitution

Solve 22.7x+13.6y+0z=408.24 for x.

$$x=-\frac{136}{227}y+\frac{20412}{1135}$$

Substitute -\frac{136}{227}y+\frac{20412}{1135} for x in the second and third equation.

$$13.6\left(-\frac{136}{227}y+\frac{20412}{1135}\right)+13.6y+22.68z=362.88 9.07\left(-\frac{136}{227}y+\frac{20412}{1135}\right)+13.6y+22.68z=317.52$$

Solve these equations for y and z respectively.

$$y=\frac{23976}{1105}-\frac{18387}{4420}z z=-\frac{46342}{128709}y+\frac{7727}{1135}$$

Substitute \frac{23976}{1105}-\frac{18387}{4420}z for y in the equation z=-\frac{46342}{128709}y+\frac{7727}{1135}.

$$z=-\frac{46342}{128709}\left(\frac{23976}{1105}-\frac{18387}{4420}z\right)+\frac{7727}{1135}$$

Solve z=-\frac{46342}{128709}\left(\frac{23976}{1105}-\frac{18387}{4420}z\right)+\frac{7727}{1135} for z.

$$z=\frac{914}{453}$$

Substitute \frac{914}{453} for z in the equation y=\frac{23976}{1105}-\frac{18387}{4420}z.

$$y=\frac{23976}{1105}-\frac{18387}{4420}\times \frac{914}{453}$$

Calculate y from y=\frac{23976}{1105}-\frac{18387}{4420}\times \frac{914}{453}.

$$y=\frac{341523}{25670}$$

Substitute \frac{341523}{25670} for y in the equation x=-\frac{136}{227}y+\frac{20412}{1135}.

$$x=-\frac{136}{227}\times \frac{341523}{25670}+\frac{20412}{1135}$$

Calculate x from x=-\frac{136}{227}\times \frac{341523}{25670}+\frac{20412}{1135}.

$$x=\frac{1512}{151}$$

The system is now solved.

$$x=\frac{1512}{151} y=\frac{341523}{25670} z=\frac{914}{453}$$