To solve the given equation [tex]\(2^{x+1} + 2^x = 3^{y+2} - 3^y\)[/tex] for integer values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex], we need to break down and analyze both sides of the equation step by step.
Let's start with the left-hand side:
[tex]\[ 2^{x+1} + 2^x \][/tex]
We can factor out a common term from this expression:
[tex]\[ 2^{x+1} + 2^x = 2 \cdot 2^x + 2^x = 2^x (2 + 1) = 3 \cdot 2^x \][/tex]
So, the left-hand side simplifies to:
[tex]\[ 3 \cdot 2^x \][/tex]
Now, let's consider the right-hand side:
[tex]\[ 3^{y+2} - 3^y \][/tex]
Similarly, we can factor out a common term from this expression:
[tex]\[ 3^{y+2} - 3^y = 3^y \cdot 3^2 - 3^y = 3^y (9 - 1) = 3^y \cdot 8 = 8 \cdot 3^y \][/tex]
So, the right-hand side simplifies to:
[tex]\[ 8 \cdot 3^y \][/tex]
Equating the simplified left-hand side to the simplified right-hand side, we have:
[tex]\[ 3 \cdot 2^x = 8 \cdot 3^y \][/tex]
To solve this, we observe that both sides of the equation must be equal. We need to find the integer values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy this equality.
From the equation [tex]\(3 \cdot 2^x = 8 \cdot 3^y\)[/tex], we can equate coefficients for similar bases.
Dividing both sides by 3:
[tex]\[ 2^x = \frac{8 \cdot 3^y}{3} \][/tex]
[tex]\[ 2^x = 8 \cdot 3^{y-1} \][/tex]
[tex]\[ 2^x = 8 \cdot 3^{y-1} \][/tex]
We're now looking for integer values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] such that this equation holds true.
After evaluating the necessary possibilities:
We find that:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = 1 \][/tex]
Thus, the integer values that satisfy the original equation are:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = 1 \][/tex]
