Let's solve the equation by substitution.
From the first equation, we can express 3^x in terms of 3^y by rearranging the equation:
3^x = 6 + 3^y
Now, substitute this expression into the second equation:
2(6 + 3^y) + 3^y = 2112 + 2(3^y) + 3^y = 2112 + 6^y + 3^y = 21
Combining like terms:9 + 9^y = 21
Subtract 9 from both sides:9^y = 12
Taking the logarithm of both sides to solve for y:log(9^y) = log(12)y = log(12)/log(9)
y ≈ 1.2618
Now, plug the value of y back into the first equation to solve for x:
3^x = 6 + 3^(1.2618)3^x = 6 + 3^1.26183^x ≈ 9.3492
Taking the logarithm of both sides to solve for x:x = log(9.3492)/log(3)x ≈ 2.0232
Therefore, the solution to the system of equations is x ≈ 2.0232 and y ≈ 1.2618.
Let's solve the equation by substitution.
From the first equation, we can express 3^x in terms of 3^y by rearranging the equation:
3^x = 6 + 3^y
Now, substitute this expression into the second equation:
2(6 + 3^y) + 3^y = 21
12 + 2(3^y) + 3^y = 21
12 + 6^y + 3^y = 21
Combining like terms:
9 + 9^y = 21
Subtract 9 from both sides:
9^y = 12
Taking the logarithm of both sides to solve for y:
log(9^y) = log(12)
y = log(12)/log(9)
y ≈ 1.2618
Now, plug the value of y back into the first equation to solve for x:
3^x = 6 + 3^(1.2618)
3^x = 6 + 3^1.2618
3^x ≈ 9.3492
Taking the logarithm of both sides to solve for x:
x = log(9.3492)/log(3)
x ≈ 2.0232
Therefore, the solution to the system of equations is x ≈ 2.0232 and y ≈ 1.2618.