To determine the number of solutions to the system of equations given by:
- \( 8x - 6y = 48 \)
- \( 8x + 6y = 48 \)
we can add both equations together to eliminate \( y \):
\[ (8x - 6y) + (8x + 6y) = 48 + 48 \]
This simplifies to:
\[ 16x = 96 \]
Dividing both sides by 16 gives:
\[ x = 6 \]
Now, we can substitute \( x = 6 \) back into one of the original equations to find \( y \). Using the first equation:
\[ 8(6) - 6y = 48 \] \[ 48 - 6y = 48 \] \[ -6y = 0 \] \[ y = 0 \]
So, we have one unique solution: \( (x, y) = (6, 0) \).
Let's analyze if the lines represented by the equations are parallel or coincide:
- The first equation can be rewritten as \( y = \frac{4}{3}x - 8 \).
- The second equation can be rewritten as \( y = -\frac{4}{3}x + 8 \).
Since the slopes of these two lines are different, the lines intersect at exactly one point.
Thus, the system of equations has one solution.