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Let a, b, and c, be nonzero real numbers such that a+b+c=0. Compute the value of

Apr 5, 2020

#1
+631
+1

Hello SpongeBobRules24!

Distribute:

(1) $$\frac{a}{b}+\frac{a}{c}+\frac{b}{a}+\frac{b}{c}+\frac{c}{a}+\frac{c}{b}$$

Combine like terms (fractions with the same denominator):

(1) $$\frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}$$

Consider:

It was given that

(2) $$a+b+c=0$$

We solve for the numerators of the three fractions in (1).

Solving:

Let us do the fraction with $$a$$ as the denominator.

$$b+c=-a$$

We substitute this into (1):

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

Our equation is now:

$$-1+\frac{a+c}{b}+\frac{a+b}{c}$$

We repeat the process of solving for the numerators of the fractions in (1) and eventually get:
$$-1+-1+-1$$

Which evaluates to $$\boxed{-3}$$

.
Apr 5, 2020
edited by AnExtremelyLongName  Apr 5, 2020