Let's solve each equation to determine which category they belong in:
-
6g + 4 - g = 0.5(10g + 8)
Simplifying gives:
\(5g + 4 = 5g + 4\)
This is true for all \(g\).
Box: Infinity many solutions -
8z + 38 = -3(-6 - 4z)
Simplifying gives:
\(8z + 38 = 18 + 12z\)
Rearranging gives:
\(-4z + 20 = 0\)
Therefore, \(z = 5\) (one solution).
Box: One solution -
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 -
2(k + 2) = 4(k + 1)
Simplifying gives:
\(2k + 4 = 4k + 4\)
Rearranging gives:
\(-2k = 0\)
Therefore, \(k = 0\) (one solution).
Box: One solution -
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(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 -
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 -
x + 3 = x - 3
Simplifying gives:
\(3 = -3\)
This is false for all \(x\).
Box: No solution -
-(-4d + 7) = -2 + 4d
Simplifying gives:
\(4d - 7 = -2 + 4d\)
Rearranging gives:
\(-7 = -2\)
This is false for all \(d\).
Box: No solution -
-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