Question

Which equation has exactly ONE solution?
Responses
A. 8+3x/2 = 13 + x
B. 3+12x/2 = 6x + 15
C. 10+15x/5 = 3x + 10
D. 4x-20/4

Answer 1

To determine which equation has exactly one solution, we can simplify each equation to see if it yields a unique value for \( x \).

**Option A:** \( \frac{8 + 3x}{2} = 13 + x \)

Multiply both sides by 2:
\[ 8 + 3x = 26 + 2x \]
Subtract \( 2x \) from both sides:
\[ 8 + 3x - 2x = 26 \]
So:
\[ x + 8 = 26 \]
Subtract 8 from both sides:
\[ x = 18 \]
This has exactly one solution.

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**Option B:** \( \frac{3 + 12x}{2} = 6x + 15 \)

Multiply both sides by 2:
\[ 3 + 12x = 12x + 30 \]
Subtract \( 12x \) from both sides:
\[ 3 = 30 \]
This is a contradiction (false statement), meaning this equation has no solutions.

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**Option C:** \( \frac{10 + 15x}{5} = 3x + 10 \)

Multiply both sides by 5:
\[ 10 + 15x = 15x + 50 \]
Subtract \( 15x \) from both sides:
\[ 10 = 50 \]
This is also a contradiction, meaning this equation has no solutions.

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**Option D:** \( \frac{4x - 20}{4} \) (Seems to be incomplete; assuming it's equal to something)

If we assume it's set equal to 0 for analysis:
\[ \frac{4x - 20}{4} = 0 \]
Multiply both sides by 4:
\[ 4x - 20 = 0 \]
Add 20 to both sides:
\[ 4x = 20 \]
Divide by 4:
\[ x = 5 \]
This equation has exactly one solution.

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From our analysis:

- **Option A** has one solution: \( x = 18 \)
- **Option D** (if set to 0) has one solution: \( x = 5 \)
- **Options B and C** have no solutions.

Thus, both **Option A and Option D** have exactly one solution, but if you only need one correct response, you could select Option A.