2. Solve for x and check.
Clear denominators, which, in this case, amounts to cross multiplying.
Transpose known terms to one side and unknown terms to the other
Factor out the x on the left.
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 win 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.
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
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 is 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 -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
and the solution checks.
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