Solve {l}{a+b+c=10}{a+3b+2c=17}{-a+2b+3c=8}

$$\left\{ \begin{array} { l } { a + b + c = 10 } \\ { a + 3 b + 2 c = 17 } \\ { - a + 2 b + 3 c = 8 } \end{array} \right.$$

Solve for a, b, c

$$a=5$$

$$b=2$$

$$c=3$$

Short Steps Using Substitution

Solve a+b+c=10 for a.

$$a=-b-c+10$$

Substitute -b-c+10 for a in the second and third equation.

$$-b-c+10+3b+2c=17 -\left(-b-c+10\right)+2b+3c=8$$

Solve these equations for b and c respectively.

$$b=-\frac{1}{2}c+\frac{7}{2} c=\frac{9}{2}-\frac{3}{4}b$$

Substitute -\frac{1}{2}c+\frac{7}{2} for b in the equation c=\frac{9}{2}-\frac{3}{4}b.

$$c=\frac{9}{2}-\frac{3}{4}\left(-\frac{1}{2}c+\frac{7}{2}\right)$$

Solve c=\frac{9}{2}-\frac{3}{4}\left(-\frac{1}{2}c+\frac{7}{2}\right) for c.

$$c=3$$

Substitute 3 for c in the equation b=-\frac{1}{2}c+\frac{7}{2}.

$$b=-\frac{1}{2}\times 3+\frac{7}{2}$$

Calculate b from b=-\frac{1}{2}\times 3+\frac{7}{2}.

$$b=2$$

Substitute 2 for b and 3 for c in the equation a=-b-c+10.

$$a=-2-3+10$$

Calculate a from a=-2-3+10.

$$a=5$$

The system is now solved.

$$a=5 b=2 c=3$$