I have 3 boxes, "One solution", "No solution", "Infinity many solutions" put them in the correct boxes

6g + 4 - g = 0.5(10g + 8)
8z + 38 = -3(-6 - 4z)
5(2 + w) = 45 + 5m
2(k + 2) = 4(k + 1)
36 - 7m = -7(m - 5)
6(c + 1) + 5 = 13 - 2 + 6c
3 + 5n = 5(n+2) - 7
x + 3 = x - 3
-(-4d + 7) = -2 + 4d
-3(v + 4) = 2v - 37

1 answer

Let's solve each equation to determine which category they belong in:

  1. 6g + 4 - g = 0.5(10g + 8)
    Simplifying gives:
    \(5g + 4 = 5g + 4\)
    This is true for all \(g\).
    Box: Infinity many solutions

  2. 8z + 38 = -3(-6 - 4z)
    Simplifying gives:
    \(8z + 38 = 18 + 12z\)
    Rearranging gives:
    \(-4z + 20 = 0\)
    Therefore, \(z = 5\) (one solution).
    Box: One solution

  3. 5(2 + w) = 45 + 5m
    Expanding gives:
    \(10 + 5w = 45 + 5m\)
    Rearranging gives:
    \(5w - 5m = 35\)
    This can have infinitely many solutions due to the relationship between \(w\) and \(m\).
    Box: Infinity many solutions

  4. 2(k + 2) = 4(k + 1)
    Simplifying gives:
    \(2k + 4 = 4k + 4\)
    Rearranging gives:
    \(-2k = 0\)
    Therefore, \(k = 0\) (one solution).
    Box: One solution

  5. 36 - 7m = -7(m - 5)
    Simplifying gives:
    \(36 - 7m = -7m + 35\)
    Rearranging gives:
    \(36 = 35\)
    This is false for all \(m\).
    Box: No solution

  6. 6(c + 1) + 5 = 13 - 2 + 6c
    Simplifying gives:
    \(6c + 6 + 5 = 11 + 6c\)
    Rearranging gives:
    \(11 = 11\)
    This is true for all \(c\).
    Box: Infinity many solutions

  7. 3 + 5n = 5(n + 2) - 7
    Simplifying gives:
    \(3 + 5n = 5n + 10 - 7\)
    Rearranging gives:
    \(3 = 3\)
    This is true for all \(n\).
    Box: Infinity many solutions

  8. x + 3 = x - 3
    Simplifying gives:
    \(3 = -3\)
    This is false for all \(x\).
    Box: No solution

  9. -(-4d + 7) = -2 + 4d
    Simplifying gives:
    \(4d - 7 = -2 + 4d\)
    Rearranging gives:
    \(-7 = -2\)
    This is false for all \(d\).
    Box: No solution

  10. -3(v + 4) = 2v - 37
    Simplifying gives:
    \(-3v - 12 = 2v - 37\)
    Combining gives:
    \(-5v = -25\)
    Therefore, \(v = 5\) (one solution).
    Box: One solution

Now, let's summarize the solutions in the correct boxes:

  • One solution:

    • 8z + 38 = -3(-6 - 4z)
    • 2(k + 2) = 4(k + 1)
    • 5(2 + w) = 45 + 5m
    • -3(v + 4) = 2v - 37
  • No solution:

    • 36 - 7m = -7(m - 5)
    • x + 3 = x - 3
    • -(-4d + 7) = -2 + 4d
  • Infinity many solutions:

    • 6g + 4 - g = 0.5(10g + 8)
    • 6(c + 1) + 5 = 13 - 2 + 6c
    • 3 + 5n = 5(n + 2) - 7