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 this system by either substitution or elimination. Here, we'll use elimination.
First, let's rewrite both equations for clarity:
- Equation 1: \( 8x - 6y = 48 \)
- Equation 2: \( 8x + 6y = 48 \)
Now, we can subtract Equation 1 from Equation 2:
\[ (8x + 6y) - (8x - 6y) = 48 - 48 \]
This simplifies to: \[ 12y = 0 \]
Solving for \( y \) gives: \[ y = 0 \]
Now, we can substitute \( y = 0 \) back into either of the original equations to find \( x \). We'll use Equation 1:
\[ 8x - 6(0) = 48 \]
This simplifies to: \[ 8x = 48 \]
Dividing both sides by 8 gives: \[ x = 6 \]
So we have found a single solution: \( (x, y) = (6, 0) \).
To summarize, the system of equations has exactly one solution.