To solve the equation \(55_x + 52_x = 77_{10}\), we need to convert the numbers from base \(x\) to base 10 and then solve for \(x\).

Let's break down the given equation step by step:

1. \(55_x\) represents a number in base \(x\) where the leftmost digit is 5 and the rightmost digit is 5.

2. \(52_x\) represents a number in base \(x\) where the leftmost digit is 5 and the rightmost digit is 2.

3. \(77_{10}\) is a number in base 10.

To convert \(55_x\) to base 10, we use the formula:

\(55_x = 5 \cdot x^1 + 5 \cdot x^0\)

To convert \(52_x\) to base 10, we use the formula:

\(52_x = 5 \cdot x^1 + 2 \cdot x^0\)

So, the equation becomes:

\(5x + 5 + 5x + 2 = 77\)

Simplify the equation:

\(10x + 7 = 77\)

Now, subtract 7 from both sides:

\(10x = 70\)

Finally, divide both sides by 10:

\(x = 7\)

So, the value of \(x\) is 7.