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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)

A

1 answer

Let's analyze each equation to determine if it has one solution, no solution, or infinitely many solutions:

  1. 2x + 3 = 2x + 7

    • Subtracting \(2x\) from both sides gives \(3 = 7\), which is false.
    • No solution (Square)

  2. 7x + 3 = 7x + 3

    • This is true for all \(x\).
    • Infinitely many solutions (Circle)

  3. 4x + 22 = 8x + 10

    • Rearranging gives \(4x = 12\), so \(x = 3\).
    • One solution (Underline)

  4. 5x + 5 = 5x + 4

    • Subtracting \(5x\) from both sides gives \(5 = 4\), which is false.
    • No solution (Square)

  5. 3x + 18 = 5x + 8

    • Rearranging gives \(2x = 10\), so \(x = 5\).
    • One solution (Underline)

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

    • Expanding gives \(12x + 24 = 12x + 24\), which is true for all \(x\).
    • Infinitely many solutions (Circle)

Summary of Results:

  • Infinitely many solutions:

    • \(7x + 3 = 7x + 3\) (Circle)
    • \(3(4x + 8) = 4(3x + 6)\) (Circle)

  • No solution:

    • \(2x + 3 = 2x + 7\) (Square)
    • \(5x + 5 = 5x + 4\) (Square)

  • One solution:

    • \(4x + 22 = 8x + 10\) (Underline)
    • \(3x + 18 = 5x + 8\) (Underline)