The problem presents four equations, and we need to determine which of these has exactly one solution. Let's analyze each equation:

1. $58x + 52 = -78x - 78$

   Simplifying, we get: $58x + 78x = -78 - 52 \implies 136x = -130 \implies x = -\frac{130}{136} = -\frac{65}{68}$ This equation has one solution, $x = -\frac{65}{68}$.

2. $58x + 52 = 78x - 78$

   Simplifying, we get: $58x - 78x = -78 - 52 \implies -20x = -130 \implies x = \frac{130}{20} = 6.5$ This equation also has one solution, $x = 6.5$.

3. $-78x + 52 = -52x - 78$

   Simplifying, we get: $-78x + 52x = -78 - 52 \implies -26x = -130 \implies x = \frac{130}{26} = 5$ This equation has one solution, $x = 5$.

4. $52x + 52 = 52x - 78$

   Simplifying, we get: $52x - 52x = -78 - 52 \implies 0 = -130$ This is a contradiction, implying no solutions for this equation.

Therefore, equations A, B, and C each have exactly one solution, while equation D has no solution. Would you like to delve into any specific details about these calculations, or do you have another question?

Related Questions:

1. What is the process to check if an equation has infinite solutions?
2. How would you solve for x in equations with fractions or decimals?
3. What steps should you follow to simplify an equation before solving?
4. Can an equation have both one solution and no solution?
5. How can graphing help to determine the number of solutions for an equation?

Tip: Always simplify equations step by step and isolate the variable to clearly see the nature of the solution (unique, none, or infinite).