SOLUTION: Find the difference if (y-x) if {{{6x + 5y + sqrt(6x + 5y) =72}}} and {{{3x - 4y + sqrt(3x - 4y) =30}}}

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Question 1209223: Find the difference if (y-x) if 6x+%2B+5y+%2B+sqrt%286x+%2B+5y%29+=72 and 3x+-+4y+%2B+sqrt%283x+-+4y%29+=30
Answer by math_tutor2020(3817) About Me  (Show Source):
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Answer: -113/13


Explanation

Let w = 6x+5y
The first equation can be turned into w%2Bsqrt%28w%29+=+72

w%2Bsqrt%28w%29+=+72
sqrt%28w%29+=+72-w
%28sqrt%28w%29%29%5E2+=+%2872-w%29%5E2
w+=+5184-144w%2Bw%5E2
w%5E2-145w%2B5184=0

Use the quadratic formula to solve. I'll skip steps.
You should arrive at the roots
w+=+64 or w+=+81

Upon checking each root back in w%2Bsqrt%28w%29+=+72, you should find that only w+=+64 is valid. The other root is extraneous.

w+=+64 turns into 6x%2B5y+=+64

A quick recap:
6x+%2B+5y+%2B+sqrt%286x+%2B+5y%29+=72 transforms into 6x%2B5y+=+64


Follow similar steps for 3x-4y%2Bsqrt%283x-4y%29+=+30 to arrive at 3x-4y+=+25

The system of equations to solve is now
system%286x%2B5y+=+64%2C3x-4y+=+25%29
There are a few ways to solve: Elimination, Substitution, Matrices

Whichever path you follow, you should end up at (x,y) = (127/13, 14/13)
Then finally we have y-x = (14/13) - (127/13) = (14-127)/13 = -113/13

-113/13 = -8.692307692307 approximately
The "692307" repeats forever