Questions: Which of the following equations have exactly one solution? Choose all answers that apply: (A) 2x-31=2x-31 (B) 2x-31=-2x-31 c. 2x+31=2x-31 (D) 2x-2=2x-31

Which of the following equations have exactly one solution? Choose all answers that apply: (A) 2x-31=2x-31 (B) 2x-31=-2x-31 c. 2x+31=2x-31 (D) 2x-2=2x-31
Transcript text: Which of the following equations have exactly one solution? Choose all answers that apply: (A) $2 x-31=2 x-31$ (B) $2 x-31=-2 x-31$ c. $2 x+31=2 x-31$ (D) $2 x-2=2 x-31$
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Solution

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Solution Steps

To determine which equations have exactly one solution, we need to analyze each equation to see if it simplifies to a form where \( x \) can be uniquely determined.

  1. For equation (A) \( 2x - 31 = 2x - 31 \), we see that both sides are identical, which means it has infinitely many solutions.
  2. For equation (B) \( 2x - 31 = -2x - 31 \), we can solve for \( x \) to see if it has a unique solution.
  3. For equation (C) \( 2x + 31 = 2x - 31 \), we can solve for \( x \) to see if it has a unique solution.
  4. For equation (D) \( 2x - 2 = 2x - 31 \), we can solve for \( x \) to see if it has a unique solution.
Solution Approach
  • Simplify each equation and solve for \( x \).
  • Check if the equation has a unique solution.
Step 1: Analyze Equation (A)

The equation \( 2x - 31 = 2x - 31 \) simplifies to \( x = x \), which is always true for any value of \( x \). Therefore, this equation has infinitely many solutions.

Step 2: Analyze Equation (B)

The equation \( 2x - 31 = -2x - 31 \) can be rearranged as follows: \[ 2x + 2x = 31 - 31 \implies 4x = 0 \implies x = 0 \] This equation has exactly one solution: \( x = 0 \).

Step 3: Analyze Equation (C)

The equation \( 2x + 31 = 2x - 31 \) simplifies to: \[ 31 = -31 \] This is a contradiction, meaning there are no solutions for this equation.

Step 4: Analyze Equation (D)

The equation \( 2x - 2 = 2x - 31 \) simplifies to: \[ -2 = -31 \] This is also a contradiction, indicating that there are no solutions for this equation.

Final Answer

The only equation that has exactly one solution is (B). Thus, the answer is \[ \boxed{B} \]

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