solve for x and y

Question

2^x+3^y is equal to 59
2^x^+^3 -3^y^+^2 is equal to 13

2 to the power of 2x+3 and 3 to the power of y+2

Answer 1

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.