Classify each equation as having No Solution, One Solution, or Infinitely Many Solutions

Question

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No Solution One Solution Infinitely Many Solutions

Options:
-4x + 4x = 0
-x = 7x - 8x
5 - x = -x
x + 4 = x + 3
9 = 3 (5x - 2)
6x - (3x + 8) = 16

Answer 1

Let's classify each of the equations based on the number of solutions they have:

1. \(-4x + 4x = 0\)
   Simplifies to \(0 = 0\) → Infinitely Many Solutions

2. \(-x = 7x - 8x\)
   Simplifies to \(-x = -x\) → Infinitely Many Solutions

3. \(5 - x = -x\)
   Adding \(x\) to both sides gives \(5 = 0\) → No Solution

4. \(x + 4 = x + 3\)
   Subtracting \(x\) from both sides gives \(4 = 3\) → No Solution

5. \(9 = 3 (5x - 2)\)
   Simplifies to \(9 = 15x - 6\), or \(15x - 6 = 9\) → \(15x = 15\), \(x = 1\) → One Solution

6. \(6x - (3x + 8) = 16\)
   Simplifies to \(6x - 3x - 8 = 16\) → \(3x - 8 = 16\), \(3x = 24\), \(x = 8\) → One Solution

Now, we can summarize the classifications:

• Infinitely Many Solutions:

  • \(-4x + 4x = 0\)
  • \(-x = 7x - 8x\)

• No Solution:

  • \(5 - x = -x\)
  • \(x + 4 = x + 3\)

• One Solution:

  • \(9 = 3 (5x - 2)\)
  • \(6x - (3x + 8) = 16\)

Now, if we place these in the correct input:

• No Solution: \(5 - x = -x\) and \(x + 4 = x + 3\)
• One Solution: \(9 = 3 (5x - 2)\) and \(6x - (3x + 8) = 16\)
• Infinitely Many Solutions: \(-4x + 4x = 0\) and \(-x = 7x - 8x\)

This gives the final classification as follows:

• Infinitely Many Solutions: \(-4x + 4x = 0\), \(-x = 7x - 8x\)
• No Solution: \(5 - x = -x\), \(x + 4 = x + 3\)
• One Solution: \(9 = 3(5x - 2)\), \(6x - (3x + 8) = 16\)