iven that $$ 3^{2 x+3} imes 9^{x}=3^{4-3 y} $$, find an expression for $$ y $$ in terms of $$ x $$.

Question

iven that $$ 3^{2 x+3} 	imes 9^{x}=3^{4-3 y} $$, find an expression for $$ y $$ in terms of $$ x $$.

UpStudy AI · Accepted Answer

To solve for $$ y $$ in the equation:

$$
3^{2x + 3} \times 9^x = 3^{4 - 3y}
$$

follow these steps:

1. **Express $$ 9^x $$ in terms of base 3:**
   
   Since $$ 9 = 3^2 $$, we can rewrite $$ 9^x $$ as $$ (3^2)^x = 3^{2x} $$.

2. **Combine the exponents on the left side:**

$$
   3^{2x + 3} \times 3^{2x} = 3^{(2x + 3) + 2x} = 3^{4x + 3}
   $$

3. **Set the exponents equal to each other:**
   
   $$
   4x + 3 = 4 - 3y
   $$

4. **Solve for $$ y $$:**

$$
   4x + 3 - 4 = -3y \
   4x - 1 = -3y \
   y = \frac{1 - 4x}{3}
   $$

So, the expression for $$ y $$ in terms of $$ x $$ is:

$$
y = \frac{1 - 4x}{3}
$$

Answer: After simplifying, y is (1 minus four x) divided by three. Thus,

y = ( 1 − 4x ) ⁄ 3
$$ y = \frac{1 - 4x}{3} $$

iven that \( 3^{2 x+3} \times 9^{x}=3^{4-3 y} \), find an expression for \( y \) in terms of \( x \).