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

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

Solve for c, a, b

$$c=\frac{2}{3}\approx 0.666666667$$

$$a = \frac{17}{9} = 1\frac{8}{9} \approx 1.888888889$$

$$b = -\frac{11}{9} = -1\frac{2}{9} \approx -1.222222222$$

Short Steps Using Substitution

Substitute a+b for c in the equation c+3b=-3.

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

Solve the second equation for a and the third equation for b.

$$a=\frac{5}{2}+\frac{1}{2}b b=-\frac{1}{4}a-\frac{3}{4}$$

Substitute \frac{5}{2}+\frac{1}{2}b for a in the equation b=-\frac{1}{4}a-\frac{3}{4}.

$$b=-\frac{1}{4}\left(\frac{5}{2}+\frac{1}{2}b\right)-\frac{3}{4}$$

Solve b=-\frac{1}{4}\left(\frac{5}{2}+\frac{1}{2}b\right)-\frac{3}{4} for b.

$$b=-\frac{11}{9}$$

Substitute -\frac{11}{9} for b in the equation a=\frac{5}{2}+\frac{1}{2}b.

$$a=\frac{5}{2}+\frac{1}{2}\left(-\frac{11}{9}\right)$$

Calculate a from a=\frac{5}{2}+\frac{1}{2}\left(-\frac{11}{9}\right).

$$a=\frac{17}{9}$$

Substitute \frac{17}{9} for a and -\frac{11}{9} for b in the equation c=a+b.

$$c=\frac{17}{9}-\frac{11}{9}$$

Calculate c from c=\frac{17}{9}-\frac{11}{9}.

$$c=\frac{2}{3}$$

The system is now solved.

$$c=\frac{2}{3} a=\frac{17}{9} b=-\frac{11}{9}$$