Solve {l}{a+c+b=2/3}{2b=a+c}{b+2c=4/9}

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

Solve for a, c, b

$$a=\frac{1}{3}\approx 0.333333333$$

$$c=\frac{1}{9}\approx 0.111111111$$

$$b=\frac{2}{9}\approx 0.222222222$$

Short Steps Using Substitution

Solve a+c+b=\frac{2}{3} for a.

$$a=-c-b+\frac{2}{3}$$

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

$$2b=-c-b+\frac{2}{3}+c$$

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

$$b=\frac{2}{9} c=-\frac{1}{2}b+\frac{2}{9}$$

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

$$c=-\frac{1}{2}\times \frac{2}{9}+\frac{2}{9}$$

Calculate c from c=-\frac{1}{2}\times \frac{2}{9}+\frac{2}{9}.

$$c=\frac{1}{9}$$

Substitute \frac{2}{9} for b and \frac{1}{9} for c in the equation a=-c-b+\frac{2}{3}.

$$a=-\frac{1}{9}-\frac{2}{9}+\frac{2}{3}$$

Calculate a from a=-\frac{1}{9}-\frac{2}{9}+\frac{2}{3}.

$$a=\frac{1}{3}$$

The system is now solved.

$$a=\frac{1}{3} c=\frac{1}{9} b=\frac{2}{9}$$