2. Solve for   x   and check.

Clear denominators, which, in this case, amounts to cross multiplying.

N(x + a) = M(x + b)

Remove parentheses

Nx + Na = Mx + Mb

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

Nx - Mx =Mb - Na

Factor out the   x   on the left.

x(N - M) =Mb - Na

Divide both sides by the coefficient of the unknown.

There is no solution if the denominator is 0. At this point we need to check and see when we will get 0 denominators. In this case, we will get a 0 denominator if   M = N.  In that case, the fraction on the right in the original equation will reduce to a 1, and so the fraction on the left will also have to reduce to a 1. That will happen if   x + a = x + b,   and that equation will have no solution unless   a = b.   If so then any value of   x   will be a solution, and there will not be a unique solution. If not then no value of   x   will be a solution and there will be no solution to the equation. Otherwise, the formula in the solution gives us our uniquely determined value of   x   which is the solution of the equation.

Check

Copy down the original equation

except where you see an  x,   copy down what   x   is equal to

The left side is a exercise like number 1. Find common denominators

Multiply out the second fractions on both the top and bottom.

Now that we have common denominators on both the top and bottom, we can add the fractions on both the top and bottom.

This simplifies to

after conveniently rearranging the terms in the top. We can factor an   M   out of the top of the top and an   N   out of the top of the bottom.

At this point we see that if   a - b = 0,   or, if   a = b,   then the solution will not check. In that case, the left side of the original equation will reduce to 1 for any value of   x   except   x =  -a =  -b,   so the right side of the original equation would also have to reduce to 1. If   M = N,   then any   x   except   x =  -a =  -b   will be a solution. Otherwise there will be no solution.

Another reason that we factor before we invert and multiply is because, as we have seen before, we can cancel at this step. Cancel the factors which would cancel if we inverted and multiplied. If we invert and multiply we get

or

and the solution checks.

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