Solve {l}{3a+2b+5c=9}{-2a-8b+c=5}{a+9b-4c=-3}

$$\left\{ \begin{array} { l } { 3 a + 2 b + 5 c = 9 } \\ { - 2 a - 8 b + c = 5 } \\ { a + 9 b - 4 c = - 3 } \end{array} \right.$$

Solve for a, b, c

$$a = \frac{346}{5} = 69\frac{1}{5} = 69.2$$

$$b = -\frac{109}{5} = -21\frac{4}{5} = -21.8$$

$$c=-31$$

Short Steps Using Substitution

Reorder the equations.

$$-2a-8b+c=5 3a+2b+5c=9 a+9b-4c=-3$$

Solve -2a-8b+c=5 for c.

$$c=2a+8b+5$$

Substitute 2a+8b+5 for c in the second and third equation.

$$3a+2b+5\left(2a+8b+5\right)=9 a+9b-4\left(2a+8b+5\right)=-3$$

Solve these equations for b and a respectively.

$$b=-\frac{13}{42}a-\frac{8}{21} a=-\frac{23}{7}b-\frac{17}{7}$$

Substitute -\frac{13}{42}a-\frac{8}{21} for b in the equation a=-\frac{23}{7}b-\frac{17}{7}.

$$a=-\frac{23}{7}\left(-\frac{13}{42}a-\frac{8}{21}\right)-\frac{17}{7}$$

Solve a=-\frac{23}{7}\left(-\frac{13}{42}a-\frac{8}{21}\right)-\frac{17}{7} for a.

$$a=\frac{346}{5}$$

Substitute \frac{346}{5} for a in the equation b=-\frac{13}{42}a-\frac{8}{21}.

$$b=-\frac{13}{42}\times \frac{346}{5}-\frac{8}{21}$$

Calculate b from b=-\frac{13}{42}\times \frac{346}{5}-\frac{8}{21}.

$$b=-\frac{109}{5}$$

Substitute -\frac{109}{5} for b and \frac{346}{5} for a in the equation c=2a+8b+5.

$$c=2\times \frac{346}{5}+8\left(-\frac{109}{5}\right)+5$$

Calculate c from c=2\times \frac{346}{5}+8\left(-\frac{109}{5}\right)+5.

$$c=-31$$

The system is now solved.

$$a=\frac{346}{5} b=-\frac{109}{5} c=-31$$