The common denominator is abc. So multiplying both sides by abc will cancel the fractions.
The result is:
xbc + yac + zab = abc
In order to solve for a, all of the terms containing A must be on one side, and
those terms not containing A on the other side.
To do this, it will be enough to move abc and xbc to their opposite sides,
changing their signs in the process (that is subtracting them from both sides)
The equation is now:
yac + zab -abc = -xbc
Factoring out the a on the left side:
a (yc + zb - bc) = -xbc
Therefore, a = -xbc/(yc + zb - bc) provided that the denominator is not zero for some combination of b,c,x,y and z
You MUST check your answer.
Now I will prove the answer is correct.
The reciprocal of a is 1/a = -(yc + zb - bc)/xbc
When this expression is multiplied by x, the cancels the x
in the denominator, and it becomes -(yc + zb - bc)/bc =
(bc - yc - zb)/bc <-- first term in the original equation
This is the first term in the original equation when the answer we found
for A is substituted. I made it italics and documented it, so we can find it later.
Now the other two terms are the fractions y/b + z/c. Their common denominator
is bc. So when added, the last two terms of the original equation becomes:
(yc + bz)/bc
This is the expression for the sum of the last two fractions in the original equation.
Notice that the expression for the first term and the last two terms both have denominators of bc.
Therefore, we can just add the numerators. Also notice that in the expression for the first term,
the signs of yc and bz are opposite. So they will cancel when added. This leaves bc/bc = 1.
The answer is PROVEN to be correct.
Yes, you can choose reasonable values for b,c,x,y and z, find what A has to be, then plug them all into
the original equation to verify the answer is 1. But that does not prove the answer is correct for ALL values.
For example, if b=c=x=y=z=1, then A = -xbc/(yc + zb - bc) = (-1)/(1+1 -1) = -1/(2 -1) = -1 / 1 = -1
Substituting these values into the original equation results in: 1/(-1) + 1/1 + 1/1 = -1 + 1 + 1 = 0 + 1 = 1