Let \( x \) be the number of board books and \( y \) be the number of soft cover books that Santiago buys. We can set up the following equation based on the costs of the books:
\[ 1.5x + 0.5y = 20 \]
To simplify this equation, we can multiply every term by 2 to eliminate the decimal:
\[ 3x + y = 40 \]
Now, we need to determine the number of non-negative integer solutions (where \( x \) and \( y \) both must be at least 0) to this equation.
Next, we can isolate \( y \):
\[ y = 40 - 3x \]
For \( y \) to be non-negative, we have:
\[ 40 - 3x \geq 0 \]
This simplifies to:
\[ 3x \leq 40 \quad \Rightarrow \quad x \leq \frac{40}{3} \quad \Rightarrow \quad x \leq 13.33 \]
Since \( x \) must be a non-negative integer, the largest integer value for \( x \) is 13. Thus, \( x \) can take any integer value from 0 to 13 (inclusive). These integers are:
\[ x = 0, 1, 2, ..., 13 \]
Now we can count how many integer solutions there are for \( x \):
- The possible values for \( x \) are \( 0, 1, 2, ..., 13 \), which gives us a total of \( 13 - 0 + 1 = 14 \) possible values.
Therefore, the number of non-negative integer solutions for the equation \( 3x + y = 40 \) is:
\[ \boxed{14} \]
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