Solve each equation for the indicated variable.

Question

(x/a)+(y/b)+(z/c)=1

 

for a.

Answer 1

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

 

Answer 2

The goal is to get the term with a by itself.

 

(x/a)+(y/b)+(z/c)=1     subtract y/b and z/c from both sides

(x/a) = 1 - (y/b) - (z/c)   on right change all terms into fractions with common denominator (bc)

(x/a) = (bc)/(bc) - (cy)/(bc) - (bz)/(bc)  combine all numerators over the common denominator

(x/a) = [ bc - cy - bz ] / (bc)  take the reciprocal of both sides

(a/x) = (bc) / [ bc - cy - bz ]   multiply both sides by x

a = (bcx) / [ bc - cy - bz ]   {solution}