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Answer :
To determine which equations have exactly one solution, let's examine each option step by step:
A) [tex]\(103x - 6 = 103x - 103\)[/tex]
- Subtract [tex]\(103x\)[/tex] from both sides:
[tex]\(-6 = -103\)[/tex]
- This results in a false statement, indicating that there is no solution.
B) [tex]\(-6x - 6 = -6x - 103\)[/tex]
- Add [tex]\(6x\)[/tex] to both sides:
[tex]\(-6 = -103\)[/tex]
- This also results in a false statement, indicating that there is no solution.
C) [tex]\(-103x - 6 = -6x - 103\)[/tex]
- Rearrange terms to bring similar terms to one side:
[tex]\(-103x + 6x = -103 + 6\)[/tex]
- Simplify to:
[tex]\(-97x = -97\)[/tex]
- Divide both sides by [tex]\(-97\)[/tex]:
[tex]\(x = 1\)[/tex]
- This equation has exactly one solution.
D) [tex]\(-6x - 6 = 103x - 103\)[/tex]
- Rearrange terms to bring similar terms to one side:
[tex]\(-6x - 103x = -103 + 6\)[/tex]
- Simplify to:
[tex]\(-109x = -97\)[/tex]
- Divide both sides by [tex]\(-109\)[/tex]:
[tex]\(x = \frac{97}{109}\)[/tex]
- This equation also has exactly one solution.
Based on the evaluation, equations C) [tex]\(-103x - 6 = -6x - 103\)[/tex] and D) [tex]\(-6x - 6 = 103x - 103\)[/tex] have exactly one solution.
A) [tex]\(103x - 6 = 103x - 103\)[/tex]
- Subtract [tex]\(103x\)[/tex] from both sides:
[tex]\(-6 = -103\)[/tex]
- This results in a false statement, indicating that there is no solution.
B) [tex]\(-6x - 6 = -6x - 103\)[/tex]
- Add [tex]\(6x\)[/tex] to both sides:
[tex]\(-6 = -103\)[/tex]
- This also results in a false statement, indicating that there is no solution.
C) [tex]\(-103x - 6 = -6x - 103\)[/tex]
- Rearrange terms to bring similar terms to one side:
[tex]\(-103x + 6x = -103 + 6\)[/tex]
- Simplify to:
[tex]\(-97x = -97\)[/tex]
- Divide both sides by [tex]\(-97\)[/tex]:
[tex]\(x = 1\)[/tex]
- This equation has exactly one solution.
D) [tex]\(-6x - 6 = 103x - 103\)[/tex]
- Rearrange terms to bring similar terms to one side:
[tex]\(-6x - 103x = -103 + 6\)[/tex]
- Simplify to:
[tex]\(-109x = -97\)[/tex]
- Divide both sides by [tex]\(-109\)[/tex]:
[tex]\(x = \frac{97}{109}\)[/tex]
- This equation also has exactly one solution.
Based on the evaluation, equations C) [tex]\(-103x - 6 = -6x - 103\)[/tex] and D) [tex]\(-6x - 6 = 103x - 103\)[/tex] have exactly one solution.
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