Solve {l}{-25+2000(a-b)+3000(a-c)=0}{2000(b-a)+5000b+5000(b-c)=0}{30000(c-a)+5000(c-b)+1000c=0}

$$\left. \begin{array} { l } { - 25 + 2000 ( a - b ) + 3000 ( a - c ) = 0 } \\ { 2000 ( b - a ) + 5000 b + 5000 ( b - c ) = 0 } \\ { 30000 ( c - a ) + 5000 ( c - b ) + 1000 c = 0 } \end{array} \right.$$

Solve for a, b, c

$$a=\frac{11}{520}\approx 0.021153846$$

$$b=\frac{3}{260}\approx 0.011538462$$

$$c=\frac{1}{52}\approx 0.019230769$$

Short Steps Using Substitution

Solve -25+2000\left(a-b\right)+3000\left(a-c\right)=0 for a.

$$a=\frac{1}{200}+\frac{2}{5}b+\frac{3}{5}c$$

Substitute \frac{1}{200}+\frac{2}{5}b+\frac{3}{5}c for a in the second and third equation.

$$2000\left(b-\left(\frac{1}{200}+\frac{2}{5}b+\frac{3}{5}c\right)\right)+5000b+5000\left(b-c\right)=0 30000\left(c-\left(\frac{1}{200}+\frac{2}{5}b+\frac{3}{5}c\right)\right)+5000\left(c-b\right)+1000c=0$$

Solve these equations for b and c respectively.

$$b=\frac{1}{1120}+\frac{31}{56}c c=\frac{1}{120}+\frac{17}{18}b$$

Substitute \frac{1}{1120}+\frac{31}{56}c for b in the equation c=\frac{1}{120}+\frac{17}{18}b.

$$c=\frac{1}{120}+\frac{17}{18}\left(\frac{1}{1120}+\frac{31}{56}c\right)$$

Solve c=\frac{1}{120}+\frac{17}{18}\left(\frac{1}{1120}+\frac{31}{56}c\right) for c.

$$c=\frac{1}{52}$$

Substitute \frac{1}{52} for c in the equation b=\frac{1}{1120}+\frac{31}{56}c.

$$b=\frac{1}{1120}+\frac{31}{56}\times \frac{1}{52}$$

Calculate b from b=\frac{1}{1120}+\frac{31}{56}\times \frac{1}{52}.

$$b=\frac{3}{260}$$

Substitute \frac{3}{260} for b and \frac{1}{52} for c in the equation a=\frac{1}{200}+\frac{2}{5}b+\frac{3}{5}c.

$$a=\frac{1}{200}+\frac{2}{5}\times \frac{3}{260}+\frac{3}{5}\times \frac{1}{52}$$

Calculate a from a=\frac{1}{200}+\frac{2}{5}\times \frac{3}{260}+\frac{3}{5}\times \frac{1}{52}.

$$a=\frac{11}{520}$$

The system is now solved.

$$a=\frac{11}{520} b=\frac{3}{260} c=\frac{1}{52}$$