2. Solve for x and check.

 

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

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.

 

 

(N-M)x=Mb-Na

 

Divide both sides by the coefficient of the unknown.

 

x=(Mb-Na)/(N-M)

 

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.

 

Check

 

Copy down the original equation

 

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

 

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

 

((Mb-Na)/(N-M)+a)/( (Mb-Na)/(N-M)+b)=M/N

 

 

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

 

((Mb-Na)/(N-M)+a(N-M)/(N-M))/ ((Mb-Na)/(N-M)+b(N-M)/(N-M))=M/N

 

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

 

((Mb-Na)/(N-M)+(aN-aM)/(N-M))/ ((Mb-Na)/(N-M)+(bN-bM)/(N-M))=M/N

 

Now that we have common denominators on both the top and

bottom, we can add the fractions on both the top and bottom.

 

((Mb-Na+aN-aM)/(N-M))/ ((Mb-Na+bN-bM)/(N-M))=M/N

 

This simplifies to

 

((Mb-aM)/(N-M))/ ((-Na+bN)/(N-M))=M/N

 

We can factor an M out of the top of the top and an N out of the

top of the bottom.

 

 

(M(b-a)/(N-M))/ (N(b-a)/(N-M))=M/N

 

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

 

 

(M(b-a)/(N-M)) ((N-M)/N(b-a))=M/N

or

 

M/N=M/N

 

and the solution checks.

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