Let's denote the number of hardcover books as \( h \) and the number of paperback books as \( p \). We have the following system of equations:
- \( h + p = 25 \) (Equation representing the total number of books)
- \( 1.50h + 0.50p = 26.50 \) (Equation representing the total cost of the books)
We can solve this system of equations step by step.
From the first equation, we can express \( h \) in terms of \( p \):
\[ h = 25 - p \]
Now, we can substitute this expression for \( h \) into the second equation:
\[ 1.50(25 - p) + 0.50p = 26.50 \]
Expanding this, we have:
\[ 37.50 - 1.50p + 0.50p = 26.50 \]
Combining like terms:
\[ 37.50 - 1.00p = 26.50 \]
Now, we subtract 37.50 from both sides:
\[ -1.00p = 26.50 - 37.50 \]
\[ -1.00p = -11.00 \]
Dividing both sides by -1, we get:
\[ p = 11 \]
So, Gillian bought 11 paperback books.
To verify, we can find the number of hardcover books using \( h = 25 - p \):
\[ h = 25 - 11 = 14 \]
Now we can check the total cost:
\[ 1.50 \times 14 + 0.50 \times 11 = 21.00 + 5.50 = 26.50 \]
The solution satisfies both equations, confirming that Gillian bought 11 paperback books.
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