Solve each equation and determine if it has one solution, no solution, or infinite solutions.
Understand the Problem
The question involves solving multi-step equations and classifying them into three categories: one solution, no solution, or infinite solutions.
Answer
- One Solution for A, B, C, D, E, F, H; G needs more info.
Answer for screen readers
- A: One Solution, ( x = -8 )
- B: One Solution, ( x = -2 )
- C: One Solution, ( x = \frac{7}{4} )
- D: One Solution, ( x = -1 )
- E: One Solution, ( x = 5 )
- F: One Solution, ( x = -\frac{19}{5} )
- G: To solve, the original B value is needed for classification.
- H: One Solution, ( x = \frac{14}{15} )
Steps to Solve
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Identify the equations We have the following equations to solve:
- A: (7x - 4x = 8 - 2x - 6x)
- B: (8x = 11 - 9(1 - x))
- C: (2(x - 2) = 3 - 2x)
- D: (3 = 14x - 2x + 15)
- E: (5 - 5(5 - 7) = 40 - 5x)
- F: (4(x - 6) = 7(2x + 2))
- G: (4(4x - B) - 32 = 4x)
- H: (2x + 12 = 4(8x - 4))
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Solve equation A Simplifying: $$3x = 8 - 2x + 6x$$ Combine like terms: $$3x = 8 + 4x$$ Rearranging gives: $$-x = 8 \implies x = -8$$ → One solution.
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Solve equation B Simplifying: $$8x = 11 - 9 + 9x$$ This leads to: $$8x = 2 + 9x$$ Rearranging gives: $$-x = 2 \implies x = -2$$ → One solution.
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Solve equation C Simplifying: $$2x - 4 = 3 - 2x$$ Combine like terms: $$4x = 7 \implies x = \frac{7}{4}$$ → One solution.
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Solve equation D Simplifying: $$3 = 12x + 15$$ Rearranging leads to: $$12x = -12 \implies x = -1$$ → One solution.
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Solve equation E Simplifying: $$5 - 5(5 - 7) = 40 - 5x$$ This simplifies to: $$5 + 10 = 40 - 5x$$ Rearranging gives: $$15 + 5x = 40 \implies 5x = 25 \implies x = 5$$ → One solution.
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Solve equation F Simplifying: $$4x - 24 = 14x + 14$$ Rearranging gives: $$-10x = 38 \implies x = -\frac{19}{5}$$ → One solution.
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Solve equation G This equation doesn't have a clearly defined constant on both sides; for clarity, we need the actual content of B to solve this.
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Solve equation H Simplifying: $$2x + 12 = 32x - 16$$ Rearranging gives: $$-30x = -28 \implies x = \frac{14}{15}$$ → One solution.
- A: One Solution, ( x = -8 )
- B: One Solution, ( x = -2 )
- C: One Solution, ( x = \frac{7}{4} )
- D: One Solution, ( x = -1 )
- E: One Solution, ( x = 5 )
- F: One Solution, ( x = -\frac{19}{5} )
- G: To solve, the original B value is needed for classification.
- H: One Solution, ( x = \frac{14}{15} )
More Information
Each equation results in one specific solution, meaning they each intersect the x-axis at just one point. G requires additional information for classification.
Tips
- Neglecting to combine like terms properly can lead to incorrect solutions.
- Forgetting to distribute terms correctly in equations can affect the outcome.
- Confusing no solution with infinite solutions often happens in equations that simplify to a true statement like (0 = 0) versus one that leads to a contradiction such as (1 = 0).
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