SOLUTION: If x and y satisfy the simultaneous equations {{{ 16^x-16^y= 64512 }}} and {{{ 4^x-4^y= 224 }}}, find the sum of x and y.

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Question 900731: If x and y satisfy the simultaneous equations +16%5Ex-16%5Ey=+64512+ and +4%5Ex-4%5Ey=+224+, find the sum of x and y.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
system%28+16%5Ex-16%5Ey=+64512%2C4%5Ex-4%5Ey=+224%29+

16%5Ex-16%5Ey=+64512, and since 16=4%5E2

%284%5E2%29%5Ex-%284%5E2%29%5Ey=64512

4%5E%282x%29-4%5E%282y%29=64512

%284%5Ex%29%5E2-%284%5Ey%29%5E2=64512

%284%5Ex-4%5Ey%29%284%5Ex%2B4%5Ey%29=64512, and since 4%5Ex-4%5Ey=+224+

224%284%5Ex%2B4%5Ey%29=64512

%284%5Ex%2B4%5Ey%29=288

Now we have the system:

system%284%5Ex%2B4%5Ey=288%2C4%5Ex-4%5Ey=+224%29+

Adding the two equations:

2%2A4%5Ex=512

4%5Ex=256

4%5Ex=4%5E4

x=4

Substituting in 4%5Ex-4%5Ey=+224

4%5E4-4%5Ey=+224

256-4%5Ey=224

-4%5Ey=-32

4%5Ey=32, since 2%5E2=4 and 2%5E5=32

%282%5E2%29%5Ey=2%5E5

2%5E%282y%29=2%5E5

2y=5

y=5%2F2

So x%2By=4%2B5%2F2=8%2F2%2B5%2F2=13%2F2=6%261%2F2

Edwin