solve for x and y

Question

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

please explain in an easier method. thanks advance

Answer 1

To solve for x and y, we can first set up a system of equations using the given information:

1. 2^x + 3^y = 59
2. 2^x + 3 - 3^y + 2 = 13

Now, let's consider the second equation:

2^x + 3 - 3^y + 2 = 13
2^x - 3^y + 5 = 13
2^x - 3^y = 8

Now, we have two equations:

1. 2^x + 3^y = 59
2. 2^x - 3^y = 8

To solve this system of equations, we can add the two equations together to eliminate the variable y:

(2^x + 3^y) + (2^x - 3^y) = 59 + 8
2 * 2^x = 67
2^(x+1) = 67
x + 1 = log2(67)
x = log2(67) - 1

Now, substitute the value of x back into the first equation to solve for y:

2^(log2(67) - 1) + 3^y = 59
67/2 + 3^y = 59
y = log3(59 - 67/2) / log3(3)

Therefore, the solutions for x and y are:
x = log2(67) - 1
y = log3(59 - 67/2) / log3(3)