The equations which have exactly one solution are:
Choice A: ![-78x+52=-52x-78](https://tex.z-dn.net/?f=-78x%2B52%3D-52x-78)
Choice C: ![58x+52=78x-78](https://tex.z-dn.net/?f=58x%2B52%3D78x-78)
Choice D: ![58x+52=-78x-78](https://tex.z-dn.net/?f=58x%2B52%3D-78x-78)
Explanation:
Choice A: ![-78x+52=-52x-78](https://tex.z-dn.net/?f=-78x%2B52%3D-52x-78)
Adding both sides by 52x, we have,
![-26x+52=-78](https://tex.z-dn.net/?f=-26x%2B52%3D-78)
Subtracting both sides by 52,
![-26x=-130](https://tex.z-dn.net/?f=-26x%3D-130)
Dividing both sides by -26,
![x=5](https://tex.z-dn.net/?f=x%3D5)
Hence, the equation
has exactly one solution.
Thus, Choice A is the correct answer.
Choice B: ![52 x+52=52 x-78](https://tex.z-dn.net/?f=52%20x%2B52%3D52%20x-78)
Subtracting both sides by 52x,
![52=-78](https://tex.z-dn.net/?f=52%3D-78)
Since, both sides are not equal.
It has no solution.
Thus, Option B is not the correct answer.
Choice C: ![58x+52=78x-78](https://tex.z-dn.net/?f=58x%2B52%3D78x-78)
Subtracting both sides by 78x,
![-20x+52=-78](https://tex.z-dn.net/?f=-20x%2B52%3D-78)
Subtracting both sides by 52,
![-20 x=-130](https://tex.z-dn.net/?f=-20%20x%3D-130)
Dividing both sides by -20,
![x=6.5](https://tex.z-dn.net/?f=x%3D6.5)
Hence, the equation
has exactly one solution.
Thus, Choice C is the correct answer.
Option D: ![58x+52=-78x-78](https://tex.z-dn.net/?f=58x%2B52%3D-78x-78)
Adding both sides by 78x,
![136x+52=-78](https://tex.z-dn.net/?f=136x%2B52%3D-78)
Subtracting both sides by 52,
![136x=-130](https://tex.z-dn.net/?f=136x%3D-130)
Dividing both sides by 136,
![x=-\frac{65}{68}\\x=-0.956](https://tex.z-dn.net/?f=x%3D-%5Cfrac%7B65%7D%7B68%7D%5C%5Cx%3D-0.956)
Hence, the equation
has exactly one solution.
Thus, Choice D is the correct answer.