To determine how many solutions the system of equations has, we can analyze the two equations:
- \( 8x - 6y = 48 \)
- \( 8x + 6y = 48 \)
We can solve the system of equations by manipulation. Let's start by adding the two equations:
\[ (8x - 6y) + (8x + 6y) = 48 + 48 \]
This simplifies to:
\[ 16x = 96 \]
Now, we can solve for \(x\):
\[ x = \frac{96}{16} = 6 \]
Next, we can substitute \(x = 6\) back into one of the original equations to find \(y\). Let's use the first equation:
\[ 8(6) - 6y = 48 \]
Simplifying that gives:
\[ 48 - 6y = 48 \]
Subtracting 48 from both sides results in:
\[ -6y = 0 \]
Thus:
\[ y = 0 \]
So we have found a unique solution:
\[ (x, y) = (6, 0) \]
Now, we can check the second equation with these values:
\[ 8(6) + 6(0) = 48 \]
This simplifies to \(48 = 48\), confirming that the solution is valid.
Since we have one unique solution for \(x\) and \(y\), the system of equations has:
one solution.