Match the description of the one variable equation with the number of solutions it will have.

Question

x - 7 = 7 - x ____________
3(x + 5) = 3x + 5 ____________
10 - x = 25 ____________
2(x + 4) = 2x + 5 + 3 _____________
Options: One Solution, Two Solutions, No Solution, Infinite Solutions

Answer 1

Let's analyze each equation to determine the number of solutions:

1. Equation: \(x - 7 = 7 - x\)

   • Rearranging gives: \(x + x = 7 + 7\) → \(2x = 14\) → \(x = 7\)
   • Number of solutions: One Solution

2. Equation: \(3(x + 5) = 3x + 5\)

   • Expanding gives: \(3x + 15 = 3x + 5\)
   • Rearranging gives: \(15 = 5\), which is false.
   • Number of solutions: No Solution

3. Equation: \(10 - x = 25\)

   • Rearranging gives: \(-x = 25 - 10\) → \(-x = 15\) → \(x = -15\)
   • Number of solutions: One Solution

4. Equation: \(2(x + 4) = 2x + 5 + 3\)

   • Expanding gives: \(2x + 8 = 2x + 8\)
   • This is always true for any \(x\).
   • Number of solutions: Infinite Solutions

Now, matching the equations with their numbers of solutions:

1. \(x - 7 = 7 - x\) → One Solution
2. \(3(x + 5) = 3x + 5\) → No Solution
3. \(10 - x = 25\) → One Solution
4. \(2(x + 4) = 2x + 5 + 3\) → Infinite Solutions

So, summarizing:

• \(x - 7 = 7 - x\) → One Solution
• \(3(x + 5) = 3x + 5\) → No Solution
• \(10 - x = 25\) → One Solution
• \(2(x + 4) = 2x + 5 + 3\) → Infinite Solutions

Answer 2

are you sure this is correct

Answer 3

Let me double-check the equations and my previous responses for accuracy:

1. Equation: \(x - 7 = 7 - x\)

   • Rearranging gives: \[ x + x = 7 + 7 \implies 2x = 14 \implies x = 7 \]
   • Number of solutions: One Solution

2. Equation: \(3(x + 5) = 3x + 5\)

   • Expanding gives: \[ 3x + 15 = 3x + 5 \]
   • Rearranging gives: \[ 15 = 5 \]
   • This is always false.
   • Number of solutions: No Solution

3. Equation: \(10 - x = 25\)

   • Rearranging gives: \[ -x = 25 - 10 \implies -x = 15 \implies x = -15 \]
   • Number of solutions: One Solution

4. Equation: \(2(x + 4) = 2x + 5 + 3\)

   • Expanding gives: \[ 2x + 8 = 2x + 8 \]
   • This is always true for any \(x\).
   • Number of solutions: Infinite Solutions

Summary:

• First equation: \(x - 7 = 7 - x\) → One Solution
• Second equation: \(3(x + 5) = 3x + 5\) → No Solution
• Third equation: \(10 - x = 25\) → One Solution
• Fourth equation: \(2(x + 4) = 2x + 5 + 3\) → Infinite Solutions

Based on the analysis above, the initial conclusion is correct. Thank you for prompting a re-evaluation. If you have further questions or need more explanations, feel free to ask!