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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
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Review the following property.
When we have solve an equation ax + by = c in terms of x and y, we can presume this equation has a unique solution under the following conditions.
1) x and y are positive integers.
2) coefficients a and b are relative primes.
3) c is not a big number.
Since condition 1) tells us that x and y are positive integers. The coefficients 4 and 3 are relative primes and the constant term c is not a big number.
Thus we can presume this equation has a unique solution.
The logical reason is as follows.
We can substitute positive integers from 1 into the variable x of the equation 3y = 13 - 4x and we should notice that 13 - 4x must be a multiple of 3 since we have 3y on the left hand side.
The unique possible value of x is 1 in order to have a positive value of y from the equation.
Then we have a unique solution x = 1 and y = 3.
Since condition 1) yields a unique solution, it is sufficient.
Condition 2)
Since condition 2) allows all real numbers, there many possibilities for solutions to this equation 4x + 3y = 13.
Since condition 2) does not yield a unique solution, it is not sufficient.
Therefore, A is the answer.
Answer: A
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
