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.

$$X=\frac{1512}{151} Y=\frac{341523}{25670} Z=\frac{914}{453}$$