Answer :
To solve for which equation has the same solution as [tex]\(-16p + 37 = 49 - 21p\)[/tex], we must first determine the solution [tex]\(p\)[/tex] for the given equation.
Starting with the equation:
[tex]\[ -16p + 37 = 49 - 21p \][/tex]
### Step 1: Move all [tex]\(p\)[/tex] terms to one side
Add [tex]\(21p\)[/tex] to both sides to collect all [tex]\(p\)[/tex] terms on the left side:
[tex]\[ -16p + 21p + 37 = 49 \][/tex]
Simplify:
[tex]\[ 5p + 37 = 49 \][/tex]
### Step 2: Isolate [tex]\(p\)[/tex]
Subtract [tex]\(37\)[/tex] from both sides:
[tex]\[ 5p = 49 - 37 \][/tex]
[tex]\[ 5p = 12 \][/tex]
Divide both sides by [tex]\(5\)[/tex]:
[tex]\[ p = \frac{12}{5} \][/tex]
[tex]\[ p = 2.4 \][/tex]
Next, we need to substitute [tex]\(p = 2.4\)[/tex] into each given equation option to check which one holds true.
### Checking Option A
[tex]\[ -55 + 12p = 5p + 16 \][/tex]
Substituting [tex]\(p = 2.4\)[/tex]:
[tex]\[ -55 + 12 \cdot 2.4 = 5 \cdot 2.4 + 16 \][/tex]
[tex]\[ -55 + 28.8 = 12 + 16 \][/tex]
[tex]\[ -26.2 \neq 28 \][/tex]
Option A is NOT valid.
### Checking Option B
[tex]\[ 2 + 1.25p = -3.75p + 10 \][/tex]
Substituting [tex]\(p = 2.4\)[/tex]:
[tex]\[ 2 + 1.25 \cdot 2.4 = -3.75 \cdot 2.4 + 10 \][/tex]
[tex]\[ 2 + 3 = -9 + 10 \][/tex]
[tex]\[ 5 \neq 1 \][/tex]
Option B is NOT valid.
### Checking Option C
[tex]\[ \frac{3}{2}p - 5 + \frac{9}{4}p = 7 - \frac{5}{4}p \][/tex]
Substituting [tex]\(p = 2.4\)[/tex]:
[tex]\[ \frac{3}{2} \cdot 2.4 - 5 + \frac{9}{4} \cdot 2.4 = 7 - \frac{5}{4} \cdot 2.4 \][/tex]
[tex]\[ 3.6 - 5 + 5.4 = 7 - 3 \][/tex]
[tex]\[ 4 = 4 \][/tex]
Option C IS valid.
### Checking Option D
[tex]\[ -14 + 6p = -9 - 6p \][/tex]
Substituting [tex]\(p = 2.4\)[/tex]:
[tex]\[ -14 + 6 \cdot 2.4 = -9 - 6 \cdot 2.4 \][/tex]
[tex]\[ -14 + 14.4 = -9 - 14.4 \][/tex]
[tex]\[ 0.4 \neq -23.4 \][/tex]
Option D is NOT valid.
### Conclusion
The equation that has the same solution as the given equation is:
[tex]\[ \boxed{\text{C}} \][/tex]
Starting with the equation:
[tex]\[ -16p + 37 = 49 - 21p \][/tex]
### Step 1: Move all [tex]\(p\)[/tex] terms to one side
Add [tex]\(21p\)[/tex] to both sides to collect all [tex]\(p\)[/tex] terms on the left side:
[tex]\[ -16p + 21p + 37 = 49 \][/tex]
Simplify:
[tex]\[ 5p + 37 = 49 \][/tex]
### Step 2: Isolate [tex]\(p\)[/tex]
Subtract [tex]\(37\)[/tex] from both sides:
[tex]\[ 5p = 49 - 37 \][/tex]
[tex]\[ 5p = 12 \][/tex]
Divide both sides by [tex]\(5\)[/tex]:
[tex]\[ p = \frac{12}{5} \][/tex]
[tex]\[ p = 2.4 \][/tex]
Next, we need to substitute [tex]\(p = 2.4\)[/tex] into each given equation option to check which one holds true.
### Checking Option A
[tex]\[ -55 + 12p = 5p + 16 \][/tex]
Substituting [tex]\(p = 2.4\)[/tex]:
[tex]\[ -55 + 12 \cdot 2.4 = 5 \cdot 2.4 + 16 \][/tex]
[tex]\[ -55 + 28.8 = 12 + 16 \][/tex]
[tex]\[ -26.2 \neq 28 \][/tex]
Option A is NOT valid.
### Checking Option B
[tex]\[ 2 + 1.25p = -3.75p + 10 \][/tex]
Substituting [tex]\(p = 2.4\)[/tex]:
[tex]\[ 2 + 1.25 \cdot 2.4 = -3.75 \cdot 2.4 + 10 \][/tex]
[tex]\[ 2 + 3 = -9 + 10 \][/tex]
[tex]\[ 5 \neq 1 \][/tex]
Option B is NOT valid.
### Checking Option C
[tex]\[ \frac{3}{2}p - 5 + \frac{9}{4}p = 7 - \frac{5}{4}p \][/tex]
Substituting [tex]\(p = 2.4\)[/tex]:
[tex]\[ \frac{3}{2} \cdot 2.4 - 5 + \frac{9}{4} \cdot 2.4 = 7 - \frac{5}{4} \cdot 2.4 \][/tex]
[tex]\[ 3.6 - 5 + 5.4 = 7 - 3 \][/tex]
[tex]\[ 4 = 4 \][/tex]
Option C IS valid.
### Checking Option D
[tex]\[ -14 + 6p = -9 - 6p \][/tex]
Substituting [tex]\(p = 2.4\)[/tex]:
[tex]\[ -14 + 6 \cdot 2.4 = -9 - 6 \cdot 2.4 \][/tex]
[tex]\[ -14 + 14.4 = -9 - 14.4 \][/tex]
[tex]\[ 0.4 \neq -23.4 \][/tex]
Option D is NOT valid.
### Conclusion
The equation that has the same solution as the given equation is:
[tex]\[ \boxed{\text{C}} \][/tex]
