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Can someone help me with this?

Can someone help me with this?-example-1

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Answer:

Choice A: 11

Explanation:

We have x² - y² = 19

x² - y² = (x + y)(x-y)

So (x + y)(x-y) = 19

Since x and y are integers as per the first statement (x, y ∈ Z), we also have x + y = integer and x-y = integer

Take (x + y)(x - y) = 19

Since 19 is prime its only factors are 1 and 19

So we have two situations which can satisfy the above
Either x + y = 1 and x -y 19 or x + y = 19 and x -y = 1

1. Let's try x + y = 1 and x - y = 19
Adding both equations together gives us 2x = 20 (y-terms cancel)

==> x = 10 ==> 10 + y = 1 or y = -9

So 2x - 9 => 2(10) - (-9) = 20 + 9 = 29

This is not one of the choices so we can go on to the next case

2 Let's take x + y = 19 and x - y = 1


Adding these two equations gives us
2x = 20, x = 10 as before


Use 10 in the equation x + y = 19

==> 10 + y = 19

==> y = 19 - 10 = 9

2x - y ==> 2(10) - 9 ==> 20 - 9 = 11

So answer is choice A: 11

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