2. Solve for   x   and check.

First clear denominators. The smallest common denominator is   Q(x - a),   so multiply both sides by   Q(x - a).

The parentheses on the left tell us that we must multiply both of the terms by   Q(x - a)

This will get rid of all the denominators. Assemble the surviving factors.

Qx + P(x - a) =Q(x - a)

Remove parentheses

Qx + Px - Pa = Qx - Qa

and we see that we can subtract   Qx   from both sides. Transpose the   Pa   to the other side of the equation.

Px = Pa - Qa

and divide both sides by the coefficient of the unknown.

Check. Copy down the original equation

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

This is not very encouraging. It would take a miracle for this to work. But we can simplify the left side of the equation. In the bottom of the first term on the left, we are subtracting fractions, so we need common denominators.

After we put the fractions in the bottom of the first term on the left over a common denominator and subtract, we see that the term simplifies to

We are now ready to invert and multiply

Factor out an   a   and cancel

This simplifies to

Amazingly we have common denominators.

or

and, amazingly, it checks. We were able to come up with the very thing that would check.

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