To rewrite the first equation \(3c + 2.5p = 31\) with one variable using the expression for \(c\), we will substitute \(c = 11 - p\) into the first equation.
Starting with the first equation:
\[ 3c + 2.5p = 31 \]
Substituting \(c\) with \(11 - p\):
\[ 3(11 - p) + 2.5p = 31 \]
Now, distribute the 3:
\[ 33 - 3p + 2.5p = 31 \]
Combine the \(p\) terms:
\[ 33 - 0.5p = 31 \]
Now, you can isolate \(-0.5p\):
\[ -0.5p = 31 - 33 \]
This simplifies to:
\[ -0.5p = -2 \]
To solve for \(p\), divide both sides by \(-0.5\):
\[ p = \frac{-2}{-0.5} = 4 \]
So, \(p = 4\).
Now you can use this value of \(p\) to find \(c\) using \(c = 11 - p\):
\[ c = 11 - 4 = 7 \]
Thus, the solutions are \(p = 4\) and \(c = 7\).