Answer :
Let's determine which of the given equations have exactly one solution. We'll analyze each equation step by step:
### Equation A: [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Combine like terms. Move all terms involving [tex]\(x\)[/tex] to one side and constant terms to the other:
[tex]\[
-6x - 103x = -103 + 6
\][/tex]
2. Simplify:
[tex]\[
-109x = -97
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-97}{-109} = \frac{97}{109}
\][/tex]
Equation A has exactly one solution: [tex]\(x = \frac{97}{109}\)[/tex].
### Equation B: [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other:
[tex]\[
103x - 103x = -103 + 6
\][/tex]
2. Simplify:
[tex]\[
0 = -97
\][/tex]
This is a contradiction, meaning there is no value of [tex]\(x\)[/tex] that will satisfy the equation. Equation B has no solution.
### Equation C: [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Combine like terms. Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other:
[tex]\[
-103x + 6x = -103 + 6
\][/tex]
2. Simplify:
[tex]\[
-97x = -97
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-97}{-97} = 1
\][/tex]
Equation C has exactly one solution: [tex]\(x = 1\)[/tex].
### Equation D: [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other:
[tex]\[
-6x + 6x = -103 + 6
\][/tex]
2. Simplify:
[tex]\[
0 = -97
\][/tex]
This is also a contradiction, meaning there is no value of [tex]\(x\)[/tex] that will satisfy the equation. Equation D has no solution.
### Conclusion
The equations that have exactly one solution are:
- Equation A with solution [tex]\(x = \frac{97}{109}\)[/tex]
- Equation C with solution [tex]\(x = 1\)[/tex]
Therefore, the equations with exactly one solution are A and C.
### Equation A: [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Combine like terms. Move all terms involving [tex]\(x\)[/tex] to one side and constant terms to the other:
[tex]\[
-6x - 103x = -103 + 6
\][/tex]
2. Simplify:
[tex]\[
-109x = -97
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-97}{-109} = \frac{97}{109}
\][/tex]
Equation A has exactly one solution: [tex]\(x = \frac{97}{109}\)[/tex].
### Equation B: [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other:
[tex]\[
103x - 103x = -103 + 6
\][/tex]
2. Simplify:
[tex]\[
0 = -97
\][/tex]
This is a contradiction, meaning there is no value of [tex]\(x\)[/tex] that will satisfy the equation. Equation B has no solution.
### Equation C: [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Combine like terms. Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other:
[tex]\[
-103x + 6x = -103 + 6
\][/tex]
2. Simplify:
[tex]\[
-97x = -97
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-97}{-97} = 1
\][/tex]
Equation C has exactly one solution: [tex]\(x = 1\)[/tex].
### Equation D: [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side and constants to the other:
[tex]\[
-6x + 6x = -103 + 6
\][/tex]
2. Simplify:
[tex]\[
0 = -97
\][/tex]
This is also a contradiction, meaning there is no value of [tex]\(x\)[/tex] that will satisfy the equation. Equation D has no solution.
### Conclusion
The equations that have exactly one solution are:
- Equation A with solution [tex]\(x = \frac{97}{109}\)[/tex]
- Equation C with solution [tex]\(x = 1\)[/tex]
Therefore, the equations with exactly one solution are A and C.
