To determine which linear equations have one solution, we need to simplify each equation to see if they are consistent and can be solved to yield a unique solution. Let's go through the equations one by one:
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5x – 1 = 3(x + 11)
- Simplifying: \( 5x - 1 = 3x + 33 \) \( 5x - 3x = 33 + 1 \) \( 2x = 34 \) \( x = 17 \)
- Result: One solution (x = 17)
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4(x – 2) + 4x = 8(x – 9)
- Simplifying: \( 4x - 8 + 4x = 8x - 72 \) \( 8x - 8 = 8x - 72 \) Subtracting \( 8x \) from both sides: \( -8 = -72 \)
- Result: This is a contradiction, so there are no solutions.
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4(x – 6) + 4 = 2(x – 3)
- Simplifying: \( 4x - 24 + 4 = 2x - 6 \) \( 4x - 20 = 2x - 6 \) \( 4x - 2x = -6 + 20 \) \( 2x = 14 \) \( x = 7 \)
- Result: One solution (x = 7)
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2(x – 4) = 5(x – 3) + 3
- Simplifying: \( 2x - 8 = 5x - 15 + 3 \) \( 2x - 8 = 5x - 12 \) \( 2x - 5x = -12 + 8 \) \( -3x = -4 \) \( x = \frac{4}{3} \)
- Result: One solution (x = 4/3)
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2(x – 1) + 3x = 5(x – 2) + 3
- Simplifying: \( 2x - 2 + 3x = 5x - 10 + 3 \) \( 5x - 2 = 5x - 7 \) Subtracting \( 5x \) from both sides: \( -2 = -7 \)
- Result: This is a contradiction, so there are no solutions.
Summary:
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Equations with one solution:
- 5x – 1 = 3(x + 11)
- 4(x – 6) + 4 = 2(x – 3)
- 2(x – 4) = 5(x – 3) + 3
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Equations with no solutions:
- 4(x – 2) + 4x = 8(x – 9)
- 2(x – 1) + 3x = 5(x – 2) + 3