Solve {l}{7/100a-1/20b-1/50c=5}{b=0.4a}{-1/50a-frac{1}{50}b+frac{8}{150}c=0.01a}

$$\left\{ \begin{array} { l } { \frac { 7 } { 100 } a - \frac { 1 } { 20 } b - \frac { 1 } { 50 } c = 5 } \\ { b = 0.4 a } \\ { - \frac { 1 } { 50 } a - \frac { 1 } { 50 } b + \frac { 8 } { 150 } c = 0.01 a } \end{array} \right.$$

Solve for a, b, c

$$a = \frac{20000}{143} = 139\frac{123}{143} \approx 139.86013986$$

$$b = \frac{8000}{143} = 55\frac{135}{143} \approx 55.944055944$$

$$c = \frac{14250}{143} = 99\frac{93}{143} \approx 99.65034965$$

Short Steps Using Substitution

Multiply each equation by the least common multiple of denominators in it. Simplify.

$$7a-5b-2c=500 b=0.4a -3a-3b+8c=1.5a$$

Reorder the equations.

$$b=0.4a 7a-5b-2c=500 -3a-3b+8c=1.5a$$

Substitute 0.4a for b in the second and third equation.

$$7a-5\times 0.4a-2c=500 -3a-3\times 0.4a+8c=1.5a$$

Solve these equations for a and c respectively.

$$a=100+\frac{2}{5}c c=\frac{57}{80}a$$

Substitute 100+\frac{2}{5}c for a in the equation c=\frac{57}{80}a.

$$c=\frac{57}{80}\left(100+\frac{2}{5}c\right)$$

Solve c=\frac{57}{80}\left(100+\frac{2}{5}c\right) for c.

$$c=\frac{14250}{143}$$

Substitute \frac{14250}{143} for c in the equation a=100+\frac{2}{5}c.

$$a=100+\frac{2}{5}\times \frac{14250}{143}$$

Calculate a from a=100+\frac{2}{5}\times \frac{14250}{143}.

$$a=\frac{20000}{143}$$

Substitute \frac{20000}{143} for a in the equation b=0.4a.

$$b=0.4\times \frac{20000}{143}$$

Calculate b from b=0.4\times \frac{20000}{143}.

$$b=\frac{8000}{143}$$

The system is now solved.

$$a=\frac{20000}{143} b=\frac{8000}{143} c=\frac{14250}{143}$$