Let's solve the given equation step by step and then verify each of the given choices to find which one has the same solution.
Given equation:
[tex]\[ -16p + 37 = 49 - 21p \][/tex]
1. First, let's move all the [tex]\( p \)[/tex] terms to one side of the equation and constants to the other side:
[tex]\[ -16p + 21p + 37 = 49 \][/tex]
[tex]\[ 5p + 37 = 49 \][/tex]
2. Subtract 37 from both sides to isolate the term with [tex]\( p \)[/tex]:
[tex]\[ 5p = 49 - 37 \][/tex]
[tex]\[ 5p = 12 \][/tex]
3. Divide both sides by 5 to solve for [tex]\( p \)[/tex]:
[tex]\[ p = \frac{12}{5} \][/tex]
[tex]\[ p = 2.4 \][/tex]
Now we need to check each of the given choices and see which one has the same solution [tex]\( p = 2.4 \)[/tex].
Choice A:
[tex]\[ -55 + 12p = 5p + 16 \][/tex]
Substitute [tex]\( p = 2.4 \)[/tex]:
[tex]\[ -55 + 12(2.4) = 5(2.4) + 16 \][/tex]
[tex]\[ -55 + 28.8 = 12 + 16 \][/tex]
[tex]\[ -26.2 \neq 28 \][/tex]
So, Choice A is incorrect.
Choice B:
[tex]\[ \frac{3}{2}p - 5 + \frac{9}{4}p = 7 - \frac{5}{4}p \][/tex]
Substitute [tex]\( p = 2.4 \)[/tex]:
[tex]\[ \frac{3}{2}(2.4) - 5 + \frac{9}{4}(2.4) = 7 - \frac{5}{4}(2.4) \][/tex]
[tex]\[ 3.6 - 5 + 5.4 = 7 - 3 \][/tex]
[tex]\[ 4 \neq 4 \][/tex]
So, Choice B is incorrect.
Choice C:
[tex]\[ -14 + 6p = -9 - 6p \][/tex]
Substitute [tex]\( p = 2.4 \)[/tex]:
[tex]\[ -14 + 6(2.4) = -9 - 6(2.4) \][/tex]
[tex]\[ -14 + 14.4 = -9 - 14.4 \][/tex]
[tex]\[ 0.4 \neq -23.4 \][/tex]
So, Choice C is incorrect.
Choice D:
[tex]\[ 2 + 1.25p = -3.75p + 10 \][/tex]
Substitute [tex]\( p = 2.4 \)[/tex]:
[tex]\[ 2 + 1.25(2.4) = -3.75(2.4) + 10 \][/tex]
[tex]\[ 2 + 3 = -9 + 10 \][/tex]
[tex]\[ 5 = 1 \][/tex]
So, Choice D is incorrect.
Since none of the given equations have [tex]\( p = 2.4 \)[/tex] as their solution, there is no correct choice among the options provided.
