Solve {l}{5=a+b+c}{14=4a-2b+c}{25=9a-3b+c}

$$\left\{ \begin{array} { l } { 5 = a + b + c } \\ { 14 = 4 a - 2 b + c } \\ { 25 = 9 a - 3 b + c } \end{array} \right.$$

Solve for a, b, c

$$a=2$$

$$b=-1$$

$$c=4$$

Short Steps Using Substitution

Solve 5=a+b+c for a.

$$a=-b+5-c$$

Substitute -b+5-c for a in the second and third equation.

$$14=4\left(-b+5-c\right)-2b+c 25=9\left(-b+5-c\right)-3b+c$$

Solve these equations for b and c respectively.

$$b=1-\frac{1}{2}c c=\frac{5}{2}-\frac{3}{2}b$$

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

$$c=\frac{5}{2}-\frac{3}{2}\left(1-\frac{1}{2}c\right)$$

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

$$c=4$$

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

$$b=1-\frac{1}{2}\times 4$$

Calculate b from b=1-\frac{1}{2}\times 4.

$$b=-1$$

Substitute -1 for b and 4 for c in the equation a=-b+5-c.

$$a=-\left(-1\right)+5-4$$

Calculate a from a=-\left(-1\right)+5-4.

$$a=2$$

The system is now solved.

$$a=2 b=-1 c=4$$