Find the solution to the following equation. $$ 3^{2 x}=9^{3-x} $$ $$ x=\frac{2}{3} $$ $$ x=\frac{9}{5} $$ $$ x=\frac{3}{2} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
3^{2x} = 9^{3 - x}
$$

**Step 1: Express Both Sides with the Same Base**

Notice that $$9$$ can be expressed as $$3^2$$. Therefore:

$$
9^{3 - x} = (3^2)^{3 - x} = 3^{2(3 - x)} = 3^{6 - 2x}
$$

Now the equation becomes:

$$
3^{2x} = 3^{6 - 2x}
$$

**Step 2: Set the Exponents Equal**

Since the bases are the same and the equation holds true, the exponents must be equal:

$$
2x = 6 - 2x
$$

**Step 3: Solve for $$x$$**

$$
2x + 2x = 6
$$
$$
4x = 6
$$
$$
x = \frac{6}{4} = \frac{3}{2}
$$

**Conclusion**

The solution to the equation is:

$$
x = \frac{3}{2}
$$

Among the provided options:

- $$ x = \frac{2}{3} $$
- $$ x = \frac{9}{5} $$
- $$ x = \frac{3}{2} $$

**The correct solution is $$ x = \frac{3}{2} $$.**
$$ x = \frac{3}{2} $$

Find the solution to the following equation. \( 3^{2 x}=9^{3-x} \) \( x=\frac{2}{3} \) \( x=\frac{9}{5} \) \( x=\frac{3}{2} \)