To determine how many solutions the system of equations has, we can analyze the equations given:
- \(8x - 6y = 48\)
- \(8x + 6y = 48\)
We can rearrange both equations for easier comparison.
From the first equation, we can express \(y\): \[ 8x - 6y = 48 \] \[ -6y = 48 - 8x \] \[ y = \frac{8x - 48}{6} \] \[ y = \frac{4x - 24}{3} \]
From the second equation: \[ 8x + 6y = 48 \] \[ 6y = 48 - 8x \] \[ y = \frac{48 - 8x}{6} \] \[ y = 8 - \frac{4x}{3} \]
Next, we'll rewrite both equations in slope-intercept or standard forms and compare their slopes.
Rearranging both to the form of \(y = mx + b\):
- \(y = \frac{4}{3}x - 8\) (for first equation)
- \(y = -\frac{4}{3}x + 8\) (for second equation)
We can see that the slopes of these two lines are different:
- First equation's slope is \(\frac{4}{3}\).
- Second equation's slope is \(-\frac{4}{3}\).
Since the slopes are different, the lines will intersect at exactly one point, meaning that the system of equations has one unique solution.
Therefore, the correct response is:
one