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Ellen's math homework involves finding out how many solutions some equations have.

Drag each equation on the left to the number of solutions it has.

Choices

5(x+1) = 25
12 + 3x + 5 = x +19 + 2x - 2
4x = 3(x + 4) - 8
2x + 10 + 6x = 8x - 7
Answers

No solution
Exactly one solution
Infinitely many solutions

1 answer

Let's analyze each equation to determine how many solutions they have.

  1. 5(x + 1) = 25

    Simplifying this equation: \[ 5x + 5 = 25 \ 5x = 20 \ x = 4 \] This equation has exactly one solution.

  2. 12 + 3x + 5 = x + 19 + 2x - 2

    Simplifying the left side: \[ 3x + 17 = x + 17 + 2x \ 3x + 17 = 3x + 17 \] This is a true statement that holds for all values of \(x\). Thus, this equation has infinitely many solutions.

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

    Simplifying this equation: \[ 4x = 3x + 12 - 8 \ 4x = 3x + 4 \ 4x - 3x = 4 \ x = 4 \] This equation has exactly one solution.

  4. 2x + 10 + 6x = 8x - 7

    Simplifying this: \[ 8x + 10 = 8x - 7 \ 10 = -7 \] This is a false statement, indicating that there are no solutions.

To summarize:

  • 5(x + 1) = 25 → Exactly one solution
  • 12 + 3x + 5 = x + 19 + 2x - 2 → Infinitely many solutions
  • 4x = 3(x + 4) - 8 → Exactly one solution
  • 2x + 10 + 6x = 8x - 7 → No solution

Final placement:

  • Exactly one solution: 5(x + 1) = 25, 4x = 3(x + 4) - 8
  • Infinitely many solutions: 12 + 3x + 5 = x + 19 + 2x - 2
  • No solution: 2x + 10 + 6x = 8x - 7