To determine whether the system of equations has no solutions, infinitely many solutions, or exactly one solution, we can analyze the two equations:
\[ -5x + 4y = -7 \quad \text{(1)} \] \[ 17x - 16y = 31 \quad \text{(2)} \]
We can use the elimination or substitution method. However, a more straightforward approach to analyze the system is to rewrite both equations in slope-intercept form (y = mx + b).
Transforming Equation (1):
\[ 4y = 5x - 7 \]
Dividing by 4: \[ y = \frac{5}{4}x - \frac{7}{4} \]
Transforming Equation (2):
\[ -16y = -17x + 31 \]
Dividing by -16: \[ y = \frac{17}{16}x - \frac{31}{16} \]
Finding the Slope:
From the first equation, the slope \(m_1 = \frac{5}{4}\).
From the second equation, the slope \(m_2 = \frac{17}{16}\).
Checking Slopes:
Since \(m_1 \neq m_2\), the two lines are not parallel, meaning they will intersect at exactly one point.
Conclusion:
The system of equations has exactly one solution.
Therefore, the answer is: One Solution
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