Answer :
To determine which equations have exactly one solution, let's analyze each equation individually:
Equation A: [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
- Add [tex]\(6x\)[/tex] to both sides: [tex]\(-6x + 6x - 6 = -6x + 6x - 103\)[/tex].
- This simplifies to [tex]\(0 = -97\)[/tex], which is not true.
Since we reached a false statement, there is no solution for Equation A.
Equation B: [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
- Add [tex]\(103x\)[/tex] and [tex]\(6x\)[/tex] to balance the equation: [tex]\(-103x + 6x = -6x + 6x - 103 + 6\)[/tex].
- Simplify to get: [tex]\(-97x = -97\)[/tex].
2. Solve for [tex]\(x\)[/tex]:
- Divide both sides by [tex]\(-97\)[/tex]: [tex]\(x = 1\)[/tex].
There is exactly one solution for Equation B, which is [tex]\(x = 1\)[/tex].
Equation C: [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Move terms involving [tex]\(x\)[/tex] to one side and constants to the other:
- Add [tex]\(6x\)[/tex] to both sides: [tex]\(-6 - 6x + 6x = 103x + 6x - 103\)[/tex].
- Simplify to get: [tex]\(-6 = -109x - 103\)[/tex].
2. Solve for [tex]\(x\)[/tex]:
- Add [tex]\(103\)[/tex] to both sides: [tex]\(97 = 109x\)[/tex].
- Divide both sides by [tex]\(-109\)[/tex]: [tex]\(x = 97/109\)[/tex].
There is exactly one solution for Equation C.
Equation D: [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
- Subtract [tex]\(103x\)[/tex] from both sides: [tex]\(103x - 103x - 6 = 103x - 103x - 103\)[/tex].
- This simplifies to [tex]\(0 = -97\)[/tex], which is not true.
Since we reached a false statement, there is no solution for Equation D.
Therefore, the equations that have exactly one solution are B and C.
Equation A: [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
- Add [tex]\(6x\)[/tex] to both sides: [tex]\(-6x + 6x - 6 = -6x + 6x - 103\)[/tex].
- This simplifies to [tex]\(0 = -97\)[/tex], which is not true.
Since we reached a false statement, there is no solution for Equation A.
Equation B: [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
- Add [tex]\(103x\)[/tex] and [tex]\(6x\)[/tex] to balance the equation: [tex]\(-103x + 6x = -6x + 6x - 103 + 6\)[/tex].
- Simplify to get: [tex]\(-97x = -97\)[/tex].
2. Solve for [tex]\(x\)[/tex]:
- Divide both sides by [tex]\(-97\)[/tex]: [tex]\(x = 1\)[/tex].
There is exactly one solution for Equation B, which is [tex]\(x = 1\)[/tex].
Equation C: [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Move terms involving [tex]\(x\)[/tex] to one side and constants to the other:
- Add [tex]\(6x\)[/tex] to both sides: [tex]\(-6 - 6x + 6x = 103x + 6x - 103\)[/tex].
- Simplify to get: [tex]\(-6 = -109x - 103\)[/tex].
2. Solve for [tex]\(x\)[/tex]:
- Add [tex]\(103\)[/tex] to both sides: [tex]\(97 = 109x\)[/tex].
- Divide both sides by [tex]\(-109\)[/tex]: [tex]\(x = 97/109\)[/tex].
There is exactly one solution for Equation C.
Equation D: [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Move all terms involving [tex]\(x\)[/tex] to one side:
- Subtract [tex]\(103x\)[/tex] from both sides: [tex]\(103x - 103x - 6 = 103x - 103x - 103\)[/tex].
- This simplifies to [tex]\(0 = -97\)[/tex], which is not true.
Since we reached a false statement, there is no solution for Equation D.
Therefore, the equations that have exactly one solution are B and C.
