To determine the number of solutions for the given system of equations, we can analyze the equations directly. The system is:
- \( 8x - 6y = 48 \) (Equation 1)
- \( 8x + 6y = 48 \) (Equation 2)
We can solve this system by using the method of elimination or substitution. Let's add the two equations together:
\[ (8x - 6y) + (8x + 6y) = 48 + 48 \]
This simplifies to:
\[ 16x = 96 \]
Now, dividing both sides by 16:
\[ x = 6 \]
Now that we have a value for \(x\), we can substitute it back into either equation to find \(y\). Let's use Equation 1:
\[ 8(6) - 6y = 48 \]
This simplifies to:
\[ 48 - 6y = 48 \]
Subtracting 48 from both sides gives:
\[ -6y = 0 \]
Dividing by -6:
\[ y = 0 \]
We found a unique solution: \( (6, 0) \).
Since we have found a specific value for both variables, the system has exactly one solution.
So, the answer is one.