Solve {l}{80.5=a(204)+b(36)+c(9)}{299=a(1290)+b(204)+c(36)}{697=a(8772)+b(1296)+c(204)}

$$\left. \begin{array} { l } { 80.5 = a ( 204 ) + b ( 36 ) + c ( 9 ) } \\ { 299 = a ( 1290 ) + b ( 204 ) + c ( 36 ) } \\ { 697 = a ( 8772 ) + b ( 1296 ) + c ( 204 ) } \end{array} \right.$$

Solve for a, b, c

$$a = -\frac{2831}{1068} = -2\frac{695}{1068} \approx -2.650749064$$

$$b = \frac{14637}{712} = 20\frac{397}{712} \approx 20.55758427$$

$$c = -\frac{1175}{89} = -13\frac{18}{89} \approx -13.202247191$$

Short Steps Using Substitution

Solve 80.5=a\times 204+b\times 36+c\times 9 for a.

$$a=-\frac{3}{17}b+\frac{161}{408}-\frac{3}{68}c$$

Substitute -\frac{3}{17}b+\frac{161}{408}-\frac{3}{68}c for a in the second and third equation.

$$299=\left(-\frac{3}{17}b+\frac{161}{408}-\frac{3}{68}c\right)\times 1290+b\times 204+c\times 36 697=\left(-\frac{3}{17}b+\frac{161}{408}-\frac{3}{68}c\right)\times 8772+b\times 1296+c\times 204$$

Solve these equations for b and c respectively.

$$b=\frac{4761}{536}-\frac{237}{268}c c=\frac{1843}{122}-\frac{84}{61}b$$

Substitute \frac{4761}{536}-\frac{237}{268}c for b in the equation c=\frac{1843}{122}-\frac{84}{61}b.

$$c=\frac{1843}{122}-\frac{84}{61}\left(\frac{4761}{536}-\frac{237}{268}c\right)$$

Solve c=\frac{1843}{122}-\frac{84}{61}\left(\frac{4761}{536}-\frac{237}{268}c\right) for c.

$$c=-\frac{1175}{89}$$

Substitute -\frac{1175}{89} for c in the equation b=\frac{4761}{536}-\frac{237}{268}c.

$$b=\frac{4761}{536}-\frac{237}{268}\left(-\frac{1175}{89}\right)$$

Calculate b from b=\frac{4761}{536}-\frac{237}{268}\left(-\frac{1175}{89}\right).

$$b=\frac{14637}{712}$$

Substitute \frac{14637}{712} for b and -\frac{1175}{89} for c in the equation a=-\frac{3}{17}b+\frac{161}{408}-\frac{3}{68}c.

$$a=-\frac{3}{17}\times \frac{14637}{712}+\frac{161}{408}-\frac{3}{68}\left(-\frac{1175}{89}\right)$$

Calculate a from a=-\frac{3}{17}\times \frac{14637}{712}+\frac{161}{408}-\frac{3}{68}\left(-\frac{1175}{89}\right).

$$a=-\frac{2831}{1068}$$

The system is now solved.

$$a=-\frac{2831}{1068} b=\frac{14637}{712} c=-\frac{1175}{89}$$