3. Solve for x and check.

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

 

Clear denominators. In this case, that amounts to cross

multiplying.

 

 

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.

 

x=(Ca-Bd)/(B-C)

 

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.

 

Check.

Copy down the original equation

 

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

 

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

 

B/(((Ca-Bd)/(B-C))+a)= C/(((Ca-Bd)/(B-C))+d)

 

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.

 

 

 

 

B/((Ca-Bd)/(B-C)+a((B-C)/(B-C)))= C/((Ca-Bd)/(B-C)+d((B-C)/(B-C)))

 

 

B/((Ca-Bd)/(B-C)+(Ba-Ca)/(B-C))= C/((Ca-Bd)/(B-C)+(Bd-Cd)/(B-C))

 

 

B/((Ca-Bd+Ba-Ca)/(B-c))=C/((Ca-Bd+Bd-Cd)/(B-C))

When we combine the like terms it simplifies down to

 

B/((Ba-Bd)/(B-C))=C/((Ca-Cd)/(B-C))

 

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.

 

 

B(B-C)/(B(a-d))=C(C-D)/(C(a-d))

 

 

and both sides reduce to the same thing.

 

(B-C)/(a-d)=(B-C)/(a-d)

 

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 tow 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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