Solve {l}{a+b+c=3}{4a+2b+c=-3}{9a+3b+c=-1}

$$\left\{ \begin{array} { l } { a + b + c = 3 } \\ { 4 a + 2 b + c = - 3 } \\ { 9 a + 3 b + c = - 1 } \end{array} \right.$$

Solve for a, b, c

$$a=4$$

$$b=-18$$

$$c=17$$

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.

$$4\left(-b-c+3\right)+2b+c=-3 9\left(-b-c+3\right)+3b+c=-1$$

Solve these equations for b and c respectively.

$$b=\frac{15}{2}-\frac{3}{2}c c=\frac{7}{2}-\frac{3}{4}b$$

Substitute \frac{15}{2}-\frac{3}{2}c for b in the equation c=\frac{7}{2}-\frac{3}{4}b.

$$c=\frac{7}{2}-\frac{3}{4}\left(\frac{15}{2}-\frac{3}{2}c\right)$$

Solve c=\frac{7}{2}-\frac{3}{4}\left(\frac{15}{2}-\frac{3}{2}c\right) for c.

$$c=17$$

Substitute 17 for c in the equation b=\frac{15}{2}-\frac{3}{2}c.

$$b=\frac{15}{2}-\frac{3}{2}\times 17$$

Calculate b from b=\frac{15}{2}-\frac{3}{2}\times 17.

$$b=-18$$

Substitute -18 for b and 17 for c in the equation a=-b-c+3.

$$a=-\left(-18\right)-17+3$$

Calculate a from a=-\left(-18\right)-17+3.

$$a=4$$

The system is now solved.

$$a=4 b=-18 c=17$$