Solve {l}{94x_{1}+3x_{2}=46}{100x_{1}+89x_{2}+2x_{3}=72}{x_{1}+x_{2}+x_{3}=1}

$$\left. \begin{array} { l } { 94 x _ { 1 } + 3 x _ { 2 } = 46 } \\ { 100 x _ { 1 } + 89 x _ { 2 } + 2 x _ { 3 } = 72 } \\ { x _ { 1 } + x _ { 2 } + x _ { 3 } = 1 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1}=\frac{316}{657}\approx 0.480974125$$

$$x_{2}=\frac{518}{1971}\approx 0.262810756$$

$$x_{3}=\frac{505}{1971}\approx 0.256215119$$

Short Steps Using Substitution

Reorder the equations.

$$x_{1}+x_{2}+x_{3}=1 100x_{1}+89x_{2}+2x_{3}=72 94x_{1}+3x_{2}=46$$

Solve x_{1}+x_{2}+x_{3}=1 for x_{1}.

$$x_{1}=-x_{2}-x_{3}+1$$

Substitute -x_{2}-x_{3}+1 for x_{1} in the second and third equation.

$$100\left(-x_{2}-x_{3}+1\right)+89x_{2}+2x_{3}=72 94\left(-x_{2}-x_{3}+1\right)+3x_{2}=46$$

Solve these equations for x_{2} and x_{3} respectively.

$$x_{2}=\frac{28}{11}-\frac{98}{11}x_{3} x_{3}=\frac{24}{47}-\frac{91}{94}x_{2}$$

Substitute \frac{28}{11}-\frac{98}{11}x_{3} for x_{2} in the equation x_{3}=\frac{24}{47}-\frac{91}{94}x_{2}.

$$x_{3}=\frac{24}{47}-\frac{91}{94}\left(\frac{28}{11}-\frac{98}{11}x_{3}\right)$$

Solve x_{3}=\frac{24}{47}-\frac{91}{94}\left(\frac{28}{11}-\frac{98}{11}x_{3}\right) for x_{3}.

$$x_{3}=\frac{505}{1971}$$

Substitute \frac{505}{1971} for x_{3} in the equation x_{2}=\frac{28}{11}-\frac{98}{11}x_{3}.

$$x_{2}=\frac{28}{11}-\frac{98}{11}\times \frac{505}{1971}$$

Calculate x_{2} from x_{2}=\frac{28}{11}-\frac{98}{11}\times \frac{505}{1971}.

$$x_{2}=\frac{518}{1971}$$

Substitute \frac{518}{1971} for x_{2} and \frac{505}{1971} for x_{3} in the equation x_{1}=-x_{2}-x_{3}+1.

$$x_{1}=-\frac{518}{1971}-\frac{505}{1971}+1$$

Calculate x_{1} from x_{1}=-\frac{518}{1971}-\frac{505}{1971}+1.

$$x_{1}=\frac{316}{657}$$

The system is now solved.

$$x_{1}=\frac{316}{657} x_{2}=\frac{518}{1971} x_{3}=\frac{505}{1971}$$