The equations which have exactly one solution are:
Choice A: [tex]-78x+52=-52x-78[/tex]
Choice C: [tex]58x+52=78x-78[/tex]
Choice D: [tex]58x+52=-78x-78[/tex]
Explanation:
Choice A: [tex]-78x+52=-52x-78[/tex]
Adding both sides by 52x, we have,
[tex]-26x+52=-78[/tex]
Subtracting both sides by 52,
[tex]-26x=-130[/tex]
Dividing both sides by -26,
[tex]x=5[/tex]
Hence, the equation [tex]-78x+52=-52x-78[/tex] has exactly one solution.
Thus, Choice A is the correct answer.
Choice B: [tex]52 x+52=52 x-78[/tex]
Subtracting both sides by 52x,
[tex]52=-78[/tex]
Since, both sides are not equal.
It has no solution.
Thus, Option B is not the correct answer.
Choice C: [tex]58x+52=78x-78[/tex]
Subtracting both sides by 78x,
[tex]-20x+52=-78[/tex]
Subtracting both sides by 52,
[tex]-20 x=-130[/tex]
Dividing both sides by -20,
[tex]x=6.5[/tex]
Hence, the equation [tex]58x+52=78x-78[/tex] has exactly one solution.
Thus, Choice C is the correct answer.
Option D: [tex]58x+52=-78x-78[/tex]
Adding both sides by 78x,
[tex]136x+52=-78[/tex]
Subtracting both sides by 52,
[tex]136x=-130[/tex]
Dividing both sides by 136,
[tex]x=-\frac{65}{68}\\x=-0.956[/tex]
Hence, the equation [tex]58x+52=-78x-78[/tex] has exactly one solution.
Thus, Choice D is the correct answer.
