To determine how many solutions the system of equations has, we can analyze the two equations:
- \( 8x - 6y = 48 \)
- \( 8x + 6y = 48 \)
We start by trying to solve the system.
Step 1: Express both equations in a simplified form.
From the first equation:
\[ 8x - 6y = 48 \quad \Rightarrow \quad 6y = 8x - 48 \quad \Rightarrow \quad y = \frac{4}{3} x - 8 \]
From the second equation:
\[ 8x + 6y = 48 \quad \Rightarrow \quad 6y = 48 - 8x \quad \Rightarrow \quad y = -\frac{4}{3} x + 8 \]
Step 2: Analyze the equations.
Now, we have:
- \( y = \frac{4}{3}x - 8 \) (from the first equation)
- \( y = -\frac{4}{3}x + 8 \) (from the second equation)
Step 3: Find solutions.
Next, we can see that the two equations have different slopes: the slope of the first line is \( \frac{4}{3} \) and the slope of the second line is \( -\frac{4}{3} \). Since they have different slopes, the lines are not parallel and will intersect at exactly one point.
Conclusion:
Thus, the system of equations has one solution. The correct response is one.