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Question 1

Examine the six different equations that follow. Determine which equations have one solution, no solution, or infinitely many solutions.

Put a circle around the two equations that have infinite solutions. Put a square around the two equations that have no solution.

Underline the two equations that have one solution.

2x + 3 = 2x + 7

7x + 3 = 7x + 3

4x + 22 = 8x + 10

5x + 5 = 5x + 4

3x + 18 = 5x + 8

3(4x + 8) = 4(3x + 6)

1 answer

Let's analyze each of the equations:

  1. 2x + 3 = 2x + 7
    Subtracting 2x from both sides:
    \(3 = 7\) (This is false)
    No solution. (Square)

  2. 7x + 3 = 7x + 3
    Both sides are equal regardless of the value of x.
    Infinitely many solutions. (Circle)

  3. 4x + 22 = 8x + 10
    Subtracting 4x from both sides:
    \(22 = 4x + 10\)
    \(12 = 4x\)
    \(x = 3\) (This has one solution)
    One solution. (Underline)

  4. 5x + 5 = 5x + 4
    Subtracting 5x from both sides:
    \(5 = 4\) (This is false)
    No solution. (Square)

  5. 3x + 18 = 5x + 8
    Rearranging gives:
    \(18 - 8 = 5x - 3x\)
    \(10 = 2x\)
    \(x = 5\) (This has one solution)
    One solution. (Underline)

  6. 3(4x + 8) = 4(3x + 6)
    Expanding both sides gives:
    \(12x + 24 = 12x + 24\)
    Both sides are equal regardless of the value of x.
    Infinitely many solutions. (Circle)

In summary:

  • Equations with infinitely many solutions:
    \(7x + 3 = 7x + 3\) (Circle)
    \(3(4x + 8) = 4(3x + 6)\) (Circle)

  • Equations with no solution:
    \(2x + 3 = 2x + 7\) (Square)
    \(5x + 5 = 5x + 4\) (Square)

  • Equations with one solution:
    \(4x + 22 = 8x + 10\) (Underline)
    \(3x + 18 = 5x + 8\) (Underline)