Solve {l}{80.5=a(204)+5(36)+c(9)}{299=a(1290)+b(204)+c(36)}{697=a(8772)+b(1296)+c(20)}

$$\left. \begin{array} { l } { 80.5 = a ( 204 ) + 5 ( 36 ) + c ( 9 ) } \\ { 299 = a ( 1290 ) + b ( 204 ) + c ( 36 ) } \\ { 697 = a ( 8772 ) + b ( 1296 ) + c ( 20 ) } \end{array} \right.$$

Solve for a, c, b

$$a=-\frac{537013}{812028}\approx -0.661323255$$

$$c = \frac{532479}{135338} = 3\frac{126465}{135338} \approx 3.934438221$$

$$b = \frac{2681463}{541352} = 4\frac{516055}{541352} \approx 4.9532707$$

Short Steps Using Substitution

Solve 80.5=a\times 204+5\times 36+c\times 9 for a.

$$a=-\frac{199}{408}-\frac{3}{68}c$$

Substitute -\frac{199}{408}-\frac{3}{68}c for a in the second and third equation.

$$299=\left(-\frac{199}{408}-\frac{3}{68}c\right)\times 1290+b\times 204+c\times 36 697=\left(-\frac{199}{408}-\frac{3}{68}c\right)\times 8772+b\times 1296+c\times 20$$

Solve these equations for c and b respectively.

$$c=-\frac{7013}{158}+\frac{2312}{237}b b=\frac{3317}{864}+\frac{367}{1296}c$$

Substitute -\frac{7013}{158}+\frac{2312}{237}b for c in the equation b=\frac{3317}{864}+\frac{367}{1296}c.

$$b=\frac{3317}{864}+\frac{367}{1296}\left(-\frac{7013}{158}+\frac{2312}{237}b\right)$$

Solve b=\frac{3317}{864}+\frac{367}{1296}\left(-\frac{7013}{158}+\frac{2312}{237}b\right) for b.

$$b=\frac{2681463}{541352}$$

Substitute \frac{2681463}{541352} for b in the equation c=-\frac{7013}{158}+\frac{2312}{237}b.

$$c=-\frac{7013}{158}+\frac{2312}{237}\times \frac{2681463}{541352}$$

Calculate c from c=-\frac{7013}{158}+\frac{2312}{237}\times \frac{2681463}{541352}.

$$c=\frac{532479}{135338}$$

Substitute \frac{532479}{135338} for c in the equation a=-\frac{199}{408}-\frac{3}{68}c.

$$a=-\frac{199}{408}-\frac{3}{68}\times \frac{532479}{135338}$$

Calculate a from a=-\frac{199}{408}-\frac{3}{68}\times \frac{532479}{135338}.

$$a=-\frac{537013}{812028}$$

The system is now solved.

$$a=-\frac{537013}{812028} c=\frac{532479}{135338} b=\frac{2681463}{541352}$$