Solve {l}{-9.7x_{1}-2.5x_{2}=14}{-21.8x_{1}-15x_{2}-x_{3}=0}{23.5x_{1}+21.5x_{2}=0}

$$\left\{ \begin{array} { l } { - 9.7 x _ { 1 } - 2.5 x _ { 2 } = 14 } \\ { - 21.8 x _ { 1 } - 15 x _ { 2 } - x _ { 3 } = 0 } \\ { 23.5 x _ { 1 } + 21.5 x _ { 2 } = 0 } \end{array} \right.$$

Solve for x_1, x_2, x_3

$$x_{1} = -\frac{215}{107} = -2\frac{1}{107} \approx -2.009345794$$

$$x_{2} = \frac{235}{107} = 2\frac{21}{107} \approx 2.196261682$$

$$x_{3} = \frac{1162}{107} = 10\frac{92}{107} \approx 10.859813084$$

Short Steps Using Substitution

Solve -9.7x_{1}-2.5x_{2}=14 for x_{1}.

$$x_{1}=-\frac{25}{97}x_{2}-\frac{140}{97}$$

Substitute -\frac{25}{97}x_{2}-\frac{140}{97} for x_{1} in the second and third equation.

$$-21.8\left(-\frac{25}{97}x_{2}-\frac{140}{97}\right)-15x_{2}-x_{3}=0 23.5\left(-\frac{25}{97}x_{2}-\frac{140}{97}\right)+21.5x_{2}=0$$

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

$$x_{3}=-\frac{910}{97}x_{2}+\frac{3052}{97} x_{2}=\frac{235}{107}$$

Substitute \frac{235}{107} for x_{2} in the equation x_{3}=-\frac{910}{97}x_{2}+\frac{3052}{97}.

$$x_{3}=-\frac{910}{97}\times \frac{235}{107}+\frac{3052}{97}$$

Calculate x_{3} from x_{3}=-\frac{910}{97}\times \frac{235}{107}+\frac{3052}{97}.

$$x_{3}=\frac{1162}{107}$$

Substitute \frac{235}{107} for x_{2} in the equation x_{1}=-\frac{25}{97}x_{2}-\frac{140}{97}.

$$x_{1}=-\frac{25}{97}\times \frac{235}{107}-\frac{140}{97}$$

Calculate x_{1} from x_{1}=-\frac{25}{97}\times \frac{235}{107}-\frac{140}{97}.

$$x_{1}=-\frac{215}{107}$$

The system is now solved.

$$x_{1}=-\frac{215}{107} x_{2}=\frac{235}{107} x_{3}=\frac{1162}{107}$$