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To visualize the idea, think of a system of linear equations as straight lines on a graph. The solution to any system of equations is one and only one point where these lines intersect. With more unknowns than equations, you will find many such solutions.
Try and work with a smaller example: 3 unknowns and 2 equations.
I thought about the question and since 20,000 is given as the amount they together spend annually, then it seems that 20,000 must be the difference in the ratios of 4:3 and 3:2. All of the ratios are met if the below is true:
40,000 = Sarah's annual income 30,000 = Mary's annual income
30,000 = Sarah's savings 20,000 = Mary's savings
Then the 20,000 spending is split equally (10,000 each)
I was wondering the same thing. I believe the correct answer should be C.
since we get a concrete number for the amount of spending between the two people, we can set up 4x+3x-20000=5x... and we get the unknown multiplier of 10000 and figure out the annual income of both people.
The problem is that you are both assuming that the quantity by which each part of the ratio is "scaled" is the same. That is not correct. In doing so, you are using an additional constraint not described in the original problem. Look at the following example.
Sara's income (S) could be 40000, but could also be 20000. If S = 20000, according to the first statement, Mary's income (M) = 3*S/4. Then, if S = 20000, M = 15000.
The red numbers are the quantities by which each part of the ratio is multiplied in order to obtain the actual incomes, spendings, and savings (Manhattan GMAT defines this quantity as the "Unknown Multiplier"). You can see that they are different. It happens, however, that in the case of S = 40000 and M = 30000 (the one you described) these quantities (your x´s) are both equal to 10000.