Solve {l}{a+b+c=3}{24a+2b+c=-3}{9a+3b+c=7}

$$\left\{ \begin{array} { l } { a + b + c = 3 } \\ { 24 a + 2 b + c = - 3 } \\ { 9 a + 3 b + c = 7 } \end{array} \right.$$

Solve for a, b, c

$$a=-\frac{8}{19}\approx -0.421052632$$

$$b = \frac{70}{19} = 3\frac{13}{19} \approx 3.684210526$$

$$c=-\frac{5}{19}\approx -0.263157895$$

Short Steps Using Substitution

Solve a+b+c=3 for a.

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

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

$$24\left(-b-c+3\right)+2b+c=-3 9\left(-b-c+3\right)+3b+c=7$$

Solve these equations for b and c respectively.

$$b=-\frac{23}{22}c+\frac{75}{22} c=\frac{5}{2}-\frac{3}{4}b$$

Substitute -\frac{23}{22}c+\frac{75}{22} for b in the equation c=\frac{5}{2}-\frac{3}{4}b.

$$c=\frac{5}{2}-\frac{3}{4}\left(-\frac{23}{22}c+\frac{75}{22}\right)$$

Solve c=\frac{5}{2}-\frac{3}{4}\left(-\frac{23}{22}c+\frac{75}{22}\right) for c.

$$c=-\frac{5}{19}$$

Substitute -\frac{5}{19} for c in the equation b=-\frac{23}{22}c+\frac{75}{22}.

$$b=-\frac{23}{22}\left(-\frac{5}{19}\right)+\frac{75}{22}$$

Calculate b from b=-\frac{23}{22}\left(-\frac{5}{19}\right)+\frac{75}{22}.

$$b=\frac{70}{19}$$

Substitute \frac{70}{19} for b and -\frac{5}{19} for c in the equation a=-b-c+3.

$$a=-\frac{70}{19}-\left(-\frac{5}{19}\right)+3$$

Calculate a from a=-\frac{70}{19}-\left(-\frac{5}{19}\right)+3.

$$a=-\frac{8}{19}$$

The system is now solved.

$$a=-\frac{8}{19} b=\frac{70}{19} c=-\frac{5}{19}$$