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We are given the following equations:
1) 2^x + 3^y = 59
2) 2^(x+3) - 3^(y+2) = 13
Let's solve these equations simultaneously:
From the second equation, we can rewrite it as (2^x)*(2^3) - (3^y)*(3^2) = 13
By simplifying this further, we get:
8*(2^x) - 9*(3^y) = 13
Now we can substitute for 2^x from the first equation:
8*(59 - 3^y) - 9*(3^y) = 13
Simplifying this, we get:
472 - 24*(3^y) - 9*(3^y) = 13
Combining like terms:
472 - 33*(3^y) = 13
33*(3^y) = 459
3^y = 459 / 33
3^y = 13
y = 2
Now substitute y = 2 into the first equation:
2^x + 3^2 = 59
2^x = 59 - 9
2^x = 50
x = log2(50) = 5.64385618977
Thus, the solution is x ≈ 5.64 and y = 2.