Let's solve the equation step by step to find the value of [tex]\( p \)[/tex].
The given equation is:
[tex]\[ -16p + 37 = 49 - 21p \][/tex]
First, we need to move all the terms involving [tex]\( p \)[/tex] to one side of the equation and the constants to the other side. Let's add [tex]\( 21p \)[/tex] to both sides of the equation:
[tex]\[ -16p + 21p + 37 = 49 \][/tex]
This simplifies to:
[tex]\[ 5p + 37 = 49 \][/tex]
Next, subtract 37 from both sides of the equation to isolate the term with [tex]\( p \)[/tex]:
[tex]\[ 5p = 49 - 37 \][/tex]
Simplifying this, we get:
[tex]\[ 5p = 12 \][/tex]
Now, divide both sides by 5 to solve for [tex]\( p \)[/tex]:
[tex]\[ p = \frac{12}{5} \][/tex]
[tex]\[ p = 2.4 \][/tex]
Therefore, the value of [tex]\( p \)[/tex] is 2.4. Now we want to find which of the given equations has the same solution.
Let's check each of the given options by substituting [tex]\( p = 2.4 \)[/tex] into them:
Option A:
[tex]\[ -55 + 12p = 5p + 16 \][/tex]
Substituting [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]
Option B:
[tex]\[ 2 + 1.25p = -3.75p + 10 \][/tex]
Substituting [tex]\( p = 2.4 \)[/tex]:
[tex]\[ 2 + 1.25(2.4) = -3.75(2.4) + 10 \][/tex]
[tex]\[ 2 + 3 = -9 + 10 \][/tex]
[tex]\[ 5 \neq 1 \][/tex]
Option C:
[tex]\[ -14 + 6p = -9 - 6p \][/tex]
Substituting [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]
Option D:
[tex]\[ \frac{3}{2}p - 5 + \frac{9}{4}p = 7 - \frac{5}{4}p \][/tex]
Substituting [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]\[ 4 = 4 \][/tex]
Thus, option D is the correct answer, as it has the same solution, [tex]\( p = 2.4 \)[/tex], as the given equation:
[tex]\[ \frac{3}{2}p - 5 + \frac{9}{4}p = 7 - \frac{5}{4}p \][/tex]
