Question 242050
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Let *[tex \Large x] represent the larger of the two positive numbers.  Let *[tex \Large y] represent the smaller.


We know their difference is 11, so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ y\ =\ 11]


We know the difference of their square roots is 1, so:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sqrt{x}\ -\ \sqrt{y}\ =\ 1]


Add *[tex \LARGE \sqrt{y}] to both sides of the second equation:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sqrt{x}\ =\ \sqrt{y}\ + 1]


Square both sides


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ y\ +\ 2\sqrt{y}\ +\ 1]


Substitute this value for *[tex \Large x] into the first equation:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (y\ +\ 2\sqrt{y}\ +\ 1)\ -\ y\ =\ 11]


Simplify:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2\sqrt{y}\ =\ 10]


And solve:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sqrt{y}\ =\ 5]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 25]


*[tex \Large y] is a positive number, so so far, so good.  Substitute this value into the first equation:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ (25)\ =\ 11]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 36]


Check:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 36\ -\ 25\ =\ 11]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 6\ -\ 5\ =\ 1]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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