Answer :
To determine which equations have exactly one solution, let's analyze each equation individually.
### Equation A: [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Start by simplifying both sides of the equation:
- The left side is [tex]\(-6x - 6\)[/tex].
- The right side is [tex]\(-6x - 103\)[/tex].
2. If we try to eliminate the [tex]\(-6x\)[/tex] terms from both sides, we get:
[tex]\(-6 = -103\)[/tex].
3. This statement is false, indicating there are no solutions for this equation.
### Equation B: [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Simplify and balance the equation:
- Move the terms involving [tex]\(x\)[/tex] to one side:
[tex]\(-103x + 6x = -103 + 6\)[/tex].
2. Simplify further:
[tex]\(-97x = -97\)[/tex].
3. Divide both sides by [tex]\(-97\)[/tex]:
[tex]\(x = 1\)[/tex].
Thus, Equation B has exactly one solution.
### Equation C: [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Simplify the equation by isolating terms involving [tex]\(x\)[/tex]:
- Rearrange terms:
[tex]\(-6x - 103x = -103 + 6\)[/tex].
2. Combine terms:
[tex]\(-109x = -97\)[/tex].
3. Divide both sides by [tex]\(-109\)[/tex]:
[tex]\(x = \frac{97}{109}\)[/tex].
Equation C has exactly one solution.
### Equation D: [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Begin simplifying by eliminating the [tex]\(103x\)[/tex] terms from each side:
[tex]\(-6 = -103\)[/tex].
2. This statement is false, indicating there are no solutions for this equation.
In conclusion, the equations that have exactly one solution are Equation B and Equation C.
### Equation A: [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Start by simplifying both sides of the equation:
- The left side is [tex]\(-6x - 6\)[/tex].
- The right side is [tex]\(-6x - 103\)[/tex].
2. If we try to eliminate the [tex]\(-6x\)[/tex] terms from both sides, we get:
[tex]\(-6 = -103\)[/tex].
3. This statement is false, indicating there are no solutions for this equation.
### Equation B: [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Simplify and balance the equation:
- Move the terms involving [tex]\(x\)[/tex] to one side:
[tex]\(-103x + 6x = -103 + 6\)[/tex].
2. Simplify further:
[tex]\(-97x = -97\)[/tex].
3. Divide both sides by [tex]\(-97\)[/tex]:
[tex]\(x = 1\)[/tex].
Thus, Equation B has exactly one solution.
### Equation C: [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Simplify the equation by isolating terms involving [tex]\(x\)[/tex]:
- Rearrange terms:
[tex]\(-6x - 103x = -103 + 6\)[/tex].
2. Combine terms:
[tex]\(-109x = -97\)[/tex].
3. Divide both sides by [tex]\(-109\)[/tex]:
[tex]\(x = \frac{97}{109}\)[/tex].
Equation C has exactly one solution.
### Equation D: [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Begin simplifying by eliminating the [tex]\(103x\)[/tex] terms from each side:
[tex]\(-6 = -103\)[/tex].
2. This statement is false, indicating there are no solutions for this equation.
In conclusion, the equations that have exactly one solution are Equation B and Equation C.
