Solve {l}{a+2b-3c=10}{-5a+b+2c=11}{7a-7b-7c=12}

$$\left. \begin{array} { l } { a + 2 b - 3 c = 10 } \\ { - 5 a + b + 2 c = 11 } \\ { 7 a - 7 b - 7 c = 12 } \end{array} \right.$$

Solve for a, b, c

$$a = -\frac{77}{17} = -4\frac{9}{17} \approx -4.529411765$$

$$b=-\frac{100}{119}\approx -0.840336134$$

$$c = -\frac{643}{119} = -5\frac{48}{119} \approx -5.403361345$$

Short Steps Using Substitution

Solve a+2b-3c=10 for a.

$$a=-2b+3c+10$$

Substitute -2b+3c+10 for a in the second and third equation.

$$-5\left(-2b+3c+10\right)+b+2c=11 7\left(-2b+3c+10\right)-7b-7c=12$$

Solve these equations for b and c respectively.

$$b=\frac{13}{11}c+\frac{61}{11} c=\frac{3}{2}b-\frac{29}{7}$$

Substitute \frac{13}{11}c+\frac{61}{11} for b in the equation c=\frac{3}{2}b-\frac{29}{7}.

$$c=\frac{3}{2}\left(\frac{13}{11}c+\frac{61}{11}\right)-\frac{29}{7}$$

Solve c=\frac{3}{2}\left(\frac{13}{11}c+\frac{61}{11}\right)-\frac{29}{7} for c.

$$c=-\frac{643}{119}$$

Substitute -\frac{643}{119} for c in the equation b=\frac{13}{11}c+\frac{61}{11}.

$$b=\frac{13}{11}\left(-\frac{643}{119}\right)+\frac{61}{11}$$

Calculate b from b=\frac{13}{11}\left(-\frac{643}{119}\right)+\frac{61}{11}.

$$b=-\frac{100}{119}$$

Substitute -\frac{100}{119} for b and -\frac{643}{119} for c in the equation a=-2b+3c+10.

$$a=-2\left(-\frac{100}{119}\right)+3\left(-\frac{643}{119}\right)+10$$

Calculate a from a=-2\left(-\frac{100}{119}\right)+3\left(-\frac{643}{119}\right)+10.

$$a=-\frac{77}{17}$$

The system is now solved.

$$a=-\frac{77}{17} b=-\frac{100}{119} c=-\frac{643}{119}$$