Answer :
To determine which equations have exactly one solution, let's analyze each option individually:
Option (A): [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Start by simplifying both sides of the equation.
The terms involving [tex]\(x\)[/tex] are the same on both sides: [tex]\(-6x\)[/tex].
2. Subtract [tex]\(-6x\)[/tex] from both sides:
[tex]\(-6 = -103\)[/tex]
This statement is false because [tex]\(-6\)[/tex] is not equal to [tex]\(-103\)[/tex], indicating that there is no solution to this equation. Thus, option A has no solution.
Option (B): [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Simplify both sides.
Here too, the [tex]\(x\)[/tex]-terms are identical on both sides: [tex]\(103x\)[/tex].
2. Subtract [tex]\(103x\)[/tex] from both sides:
[tex]\(-6 = -103\)[/tex]
Again, this is a false statement, indicating that this equation has no solution. Therefore, option B has no solution.
Option (C): [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Start by moving all terms involving [tex]\(x\)[/tex] to one side. Add [tex]\(6x\)[/tex] to both sides:
[tex]\(-6 = 103x + 6x - 103\)[/tex]
Which simplifies to:
[tex]\(-6 = 109x - 103\)[/tex]
2. Add 103 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\(97 = 109x\)[/tex]
3. Divide each side by 109:
[tex]\(x = \frac{97}{109}\)[/tex]
This equation gives a unique solution, which means there is exactly one solution for option C.
Option (D): [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Rearrange terms to put like terms on one side. Add [tex]\(103x\)[/tex] to both sides:
[tex]\(-6 = 97x - 103\)[/tex]
2. Add 103 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\(97 = 97x\)[/tex]
3. Divide each side by 97:
[tex]\(x = 1\)[/tex]
This equation also has a unique solution, giving exactly one solution for option D.
Therefore, the equations that have exactly one solution are in options (C) and (D).
Option (A): [tex]\(-6x - 6 = -6x - 103\)[/tex]
1. Start by simplifying both sides of the equation.
The terms involving [tex]\(x\)[/tex] are the same on both sides: [tex]\(-6x\)[/tex].
2. Subtract [tex]\(-6x\)[/tex] from both sides:
[tex]\(-6 = -103\)[/tex]
This statement is false because [tex]\(-6\)[/tex] is not equal to [tex]\(-103\)[/tex], indicating that there is no solution to this equation. Thus, option A has no solution.
Option (B): [tex]\(103x - 6 = 103x - 103\)[/tex]
1. Simplify both sides.
Here too, the [tex]\(x\)[/tex]-terms are identical on both sides: [tex]\(103x\)[/tex].
2. Subtract [tex]\(103x\)[/tex] from both sides:
[tex]\(-6 = -103\)[/tex]
Again, this is a false statement, indicating that this equation has no solution. Therefore, option B has no solution.
Option (C): [tex]\(-6x - 6 = 103x - 103\)[/tex]
1. Start by moving all terms involving [tex]\(x\)[/tex] to one side. Add [tex]\(6x\)[/tex] to both sides:
[tex]\(-6 = 103x + 6x - 103\)[/tex]
Which simplifies to:
[tex]\(-6 = 109x - 103\)[/tex]
2. Add 103 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\(97 = 109x\)[/tex]
3. Divide each side by 109:
[tex]\(x = \frac{97}{109}\)[/tex]
This equation gives a unique solution, which means there is exactly one solution for option C.
Option (D): [tex]\(-103x - 6 = -6x - 103\)[/tex]
1. Rearrange terms to put like terms on one side. Add [tex]\(103x\)[/tex] to both sides:
[tex]\(-6 = 97x - 103\)[/tex]
2. Add 103 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\(97 = 97x\)[/tex]
3. Divide each side by 97:
[tex]\(x = 1\)[/tex]
This equation also has a unique solution, giving exactly one solution for option D.
Therefore, the equations that have exactly one solution are in options (C) and (D).
