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

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

Solve for a, b, c

$$a = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

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

$$c=1$$

Short Steps Using Substitution

Solve a+b+c=3 for a.

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

Substitute -b-c+3 for a in the equation a-2b+4c=5.

$$-b-c+3-2b+4c=5$$

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

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

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

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

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

$$c=1$$

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

$$b=1-\frac{2}{3}$$

Calculate b from b=1-\frac{2}{3}.

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

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

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

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

$$a=\frac{5}{3}$$

The system is now solved.

$$a=\frac{5}{3} b=\frac{1}{3} c=1$$