To determine how many solutions the system of equations has, we can analyze the two equations:
- \( 8x - 6y = 48 \)
- \( 8x + 6y = 48 \)
First, let's manipulate these equations a bit.
From the first equation, we can express \( y \) in terms of \( x \):
\[ 8x - 6y = 48 \implies -6y = 48 - 8x \implies y = \frac{8x - 48}{6} \]
From the second equation, we can do the same:
\[ 8x + 6y = 48 \implies 6y = 48 - 8x \implies y = \frac{48 - 8x}{6} \]
Now, we can see if there is a relationship between the two equations. If we compare the two expressions for \( y \), we have:
- \( y = \frac{8x - 48}{6} \)
- \( y = \frac{48 - 8x}{6} \)
Setting them equal to each other:
\[ \frac{8x - 48}{6} = \frac{48 - 8x}{6} \]
We can eliminate the denominators:
\[ 8x - 48 = 48 - 8x \]
Now, rearranging gives:
\[ 8x + 8x = 48 + 48 \] \[ 16x = 96 \implies x = 6 \]
Substituting \( x = 6 \) into either equation to solve for \( y \):
Using the first equation \( 8x - 6y = 48 \):
\[ 8(6) - 6y = 48 \] \[ 48 - 6y = 48 \]
This implies \( -6y = 0 \) or \( y = 0 \).
So the solution to the system of equations is:
\[ (x, y) = (6, 0) \]
Since we found exactly one solution, the answer is:
one.