To determine the number of solutions for the given system of equations, we can analyze the two equations:
- \(3x - 8y = 4\) (Equation 1)
- \(6x - 16y = 17\) (Equation 2)
First, notice that Equation 2 can be simplified. We can rewrite it by dividing every term by 2:
\[ 3x - 8y = \frac{17}{2} \]
Now we have:
- \(3x - 8y = 4\) (Equation 1)
- \(3x - 8y = \frac{17}{2}\) (Simplified Equation 2)
We can see that both equations represent lines in the \(xy\)-plane. Since both have the same left-hand side, they must be equal to each other:
\[ 4 = \frac{17}{2} \]
This equation is not true because \(4\) (which is \(\frac{8}{2}\)) is not equal to \(\frac{17}{2}\).
Since the left-hand sides of both equations are identical but their right-hand sides are different, the two lines represented by these equations are parallel and will never intersect.
Thus, the system of equations has no solutions.