What is the value of xy? (1) y = x + 1 (2) y = x^2 + 1

We need a single value of \(xy\) in order to conclude a statement sufficient

(1) \(y = x + 1\) - Insufficient
1. Replacing \(y\) with \(x + 1\) into \(xy\) we get \(x(x+1)\)
2. But, this statement does not help us arrive at a unique value for \(xy\)

(2) \(y = x^2 + 1\) - Insufficient
1. Replacing \(y\) with \(x^2 + 1\) into \(xy\) we get \(x(x^2+1)\)
2. But, this statement does not help us arrive at a unique value for \(xy\)

\(Combined\) - Insufficient
1. Putting the values of \(y\) on either side we get, \(x + 1 = x^2 + 1\)
2. \(x^2 - x = 0\)
3. \(x(x - 1) = 0\), we get \(x = 0\) or \(x = 1\)
4. Substituting the value of \(x\) we get \(y = 1\) @ \(x = 0\) and \(y = 2\) @ \(x = 1\)
5. Using the two sets of values we can get two different values of \(xy = 0\) and \(2\)

Ans. E

Target question: What is the value of xy?

Statement 1: y = x + 1
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 0 and y = 1. In this case, the answer to the target question is xy = (0)(1) = 0
Case b: x = 1 and y = 2. In this case, the answer to the target question is xy = (1)(2) = 2
Since we can’t answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y = x² + 1
The statement doesn't feel sufficient either, so let's test some more values.
Strategy: When considering possible values to test for statement 2, you can save yourself some time by seeing if any of the values you used when testing statement 1 also work for statement 2.
In this case, BOTH pairs of values that satisfied statement 1 also satisfy statement 2, which means we can repurpose them to get...
Case a: x = 0 and y = 1. In this case, the answer to the target question is xy = (0)(1) = 0
Case b: x = 1 and y = 2. In this case, the answer to the target question is xy = (1)(2) = 2
Since we can’t answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: x = 0 and y = 1. In this case, the answer to the target question is xy = (0)(1) = 0
Case b: x = 1 and y = 2. In this case, the answer to the target question is xy = (1)(2) = 2
Since we can’t answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Important: Some students will conclude that since the combined statements include 2 different equations with 2 variables, the combined statement should be sufficient. However, this rule applies only to situations in which the two equations are LINEAR equations.

Cheers,
Brent

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