Solve {l}{285a+45b+9c=74}{2025a+285b+452=421}{15333a+2025b+2856=2770}

$$\left. \begin{array} { l } { 285 a + 45 b + 9 c = 74 } \\ { 2025 a + 285 b + 452 = 421 } \\ { 15333 a + 2025 b + 2856 = 2770 } \end{array} \right.$$

Solve for a, b, c

$$a=\frac{2551}{17952}\approx 0.142101159$$

$$b = -\frac{100391}{89760} = -1\frac{10631}{89760} \approx -1.118438057$$

$$c = \frac{11401}{1224} = 9\frac{385}{1224} \approx 9.314542484$$

Short Steps Using Substitution

Solve 285a+45b+9c=74 for a.

$$a=-\frac{3}{19}b-\frac{3}{95}c+\frac{74}{285}$$

Substitute -\frac{3}{19}b-\frac{3}{95}c+\frac{74}{285} for a in the second and third equation.

$$2025\left(-\frac{3}{19}b-\frac{3}{95}c+\frac{74}{285}\right)+285b+452=421 15333\left(-\frac{3}{19}b-\frac{3}{95}c+\frac{74}{285}\right)+2025b+2856=2770$$

Solve these equations for b and c respectively.

$$b=\frac{10579}{660}-\frac{81}{44}c c=\frac{20336}{2421}-\frac{220}{269}b$$

Substitute \frac{10579}{660}-\frac{81}{44}c for b in the equation c=\frac{20336}{2421}-\frac{220}{269}b.

$$c=\frac{20336}{2421}-\frac{220}{269}\left(\frac{10579}{660}-\frac{81}{44}c\right)$$

Solve c=\frac{20336}{2421}-\frac{220}{269}\left(\frac{10579}{660}-\frac{81}{44}c\right) for c.

$$c=\frac{11401}{1224}$$

Substitute \frac{11401}{1224} for c in the equation b=\frac{10579}{660}-\frac{81}{44}c.

$$b=\frac{10579}{660}-\frac{81}{44}\times \frac{11401}{1224}$$

Calculate b from b=\frac{10579}{660}-\frac{81}{44}\times \frac{11401}{1224}.

$$b=-\frac{100391}{89760}$$

Substitute -\frac{100391}{89760} for b and \frac{11401}{1224} for c in the equation a=-\frac{3}{19}b-\frac{3}{95}c+\frac{74}{285}.

$$a=-\frac{3}{19}\left(-\frac{100391}{89760}\right)-\frac{3}{95}\times \frac{11401}{1224}+\frac{74}{285}$$

Calculate a from a=-\frac{3}{19}\left(-\frac{100391}{89760}\right)-\frac{3}{95}\times \frac{11401}{1224}+\frac{74}{285}.

$$a=\frac{2551}{17952}$$

The system is now solved.

$$a=\frac{2551}{17952} b=-\frac{100391}{89760} c=\frac{11401}{1224}$$