3. Solve for   x   and check.

Clear denominators. In this case, that amounts to cross multiplying.

B(x + d) = C(x + a)

Remove parentheses.

Bx + BD = Cx + Ca

Transpose known terms to one side and unknown terms to the other side.

Bx - Cx = Ca - Bd

Combining like terms is accomplished by factoring out the   x.

(B - C)x = Ca - Bd

This gets all of the x's together in one place with only one thing happening to it. We can get the   x   all by itself by simply dividing both sides by the coefficient of the unknown.

Note that this expression will have a zero denominator if   B = C.   In that case, the tops of both sides of the original equation will be the same, and two fractions with the same tops are equal if and only if the denominators are equal. If we consider what it would mean if the denominators are the same,   x + a = x + d,   we get an equation which has no solution unless   a = d.   In that case, any   x   will be a solution.


Copy down the original equation

except that wherever you see an   x,   copy down the solution instead.

We get compound fractions on both sides of the equations. The expressions on the two sides do not look very much alike, but we should see what happens if we simplify both sides. Find common denominators for the two terms on each bottom.

When we combine the like terms it simplifies down to

We can now factor out common factors from the tops of the bottoms.

The reason we factor at this point is because we can cancel here. Cancel the factors that would cancel if you inverted and multiplied.

and both sides reduce to the same thing.

Note that his expression will have a zero denominator if   a = d.   In that case, the original equation would have the same denominator on both sides. The only way that two fractions that have the same denominator can be equal is if the numerators are also the same, i.e., if   B = C   which we have already discussed.

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