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Answer :
To determine which of the given equations has exactly one solution, let's analyze each equation:
Equation (A): [tex]\( 103x - 6 = 103x - 103 \)[/tex]
- Simplify both sides:
The terms [tex]\( 103x \)[/tex] on both sides cancel out, leaving [tex]\( -6 = -103 \)[/tex].
This statement is false.
Therefore, Equation (A) has no solution.
Equation (B): [tex]\( -6x - 6 = -6x - 103 \)[/tex]
- Simplify both sides:
The terms [tex]\( -6x \)[/tex] on both sides cancel out, leaving [tex]\( -6 = -103 \)[/tex].
This statement is false.
Therefore, Equation (B) has no solution.
Equation (C): [tex]\( -103x - 6 = -6x - 103 \)[/tex]
- Rearrange the equation to isolate terms with [tex]\( x \)[/tex]:
Add [tex]\( 103x \)[/tex] to both sides:
[tex]\( -6 = 97x - 103 \)[/tex].
- Solve for [tex]\( x \)[/tex]:
Add 103 to both sides:
[tex]\( 97 = 97x \)[/tex].
Divide both sides by 97:
[tex]\( x = 1 \)[/tex].
This is a true statement and provides one solution.
Therefore, Equation (C) has exactly one solution.
Equation (D): [tex]\( -6x - 6 = 103x - 103 \)[/tex]
- Rearrange to isolate terms with [tex]\( x \)[/tex]:
Add [tex]\( 6x \)[/tex] to both sides:
[tex]\(-6 = 109x - 103\)[/tex].
- Solve for [tex]\( x \)[/tex]:
Add 103 to both sides:
[tex]\( 97 = 109x \)[/tex].
Divide both sides by 109:
[tex]\( x = \frac{97}{109} \)[/tex].
This is a valid solution that simplifies down to a single value.
Therefore, Equation (D) has exactly one solution.
In conclusion, the equations that have exactly one solution are (C) and (D).
Equation (A): [tex]\( 103x - 6 = 103x - 103 \)[/tex]
- Simplify both sides:
The terms [tex]\( 103x \)[/tex] on both sides cancel out, leaving [tex]\( -6 = -103 \)[/tex].
This statement is false.
Therefore, Equation (A) has no solution.
Equation (B): [tex]\( -6x - 6 = -6x - 103 \)[/tex]
- Simplify both sides:
The terms [tex]\( -6x \)[/tex] on both sides cancel out, leaving [tex]\( -6 = -103 \)[/tex].
This statement is false.
Therefore, Equation (B) has no solution.
Equation (C): [tex]\( -103x - 6 = -6x - 103 \)[/tex]
- Rearrange the equation to isolate terms with [tex]\( x \)[/tex]:
Add [tex]\( 103x \)[/tex] to both sides:
[tex]\( -6 = 97x - 103 \)[/tex].
- Solve for [tex]\( x \)[/tex]:
Add 103 to both sides:
[tex]\( 97 = 97x \)[/tex].
Divide both sides by 97:
[tex]\( x = 1 \)[/tex].
This is a true statement and provides one solution.
Therefore, Equation (C) has exactly one solution.
Equation (D): [tex]\( -6x - 6 = 103x - 103 \)[/tex]
- Rearrange to isolate terms with [tex]\( x \)[/tex]:
Add [tex]\( 6x \)[/tex] to both sides:
[tex]\(-6 = 109x - 103\)[/tex].
- Solve for [tex]\( x \)[/tex]:
Add 103 to both sides:
[tex]\( 97 = 109x \)[/tex].
Divide both sides by 109:
[tex]\( x = \frac{97}{109} \)[/tex].
This is a valid solution that simplifies down to a single value.
Therefore, Equation (D) has exactly one solution.
In conclusion, the equations that have exactly one solution are (C) and (D).
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