To determine whether the given system of equations has infinitely many solutions, we first need to find the equations of the two lines given their respective points.
Finding the equation of the first line:
The first line passes through the points \((2, -13)\) and \((-2, -1)\).
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Calculate the slope (m) using the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1) = (2, -13)\) and \((x_2, y_2) = (-2, -1)\): \[ m = \frac{-1 - (-13)}{-2 - 2} = \frac{-1 + 13}{-4} = \frac{12}{-4} = -3 \]
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Use the point-slope form of the line equation: \[ y - y_1 = m(x - x_1) \] Using point \((2, -13)\): \[ y - (-13) = -3(x - 2) \] \[ y + 13 = -3x + 6 \] \[ y = -3x + 6 - 13 \] \[ y = -3x - 7 \]
Finding the equation of the second line:
The second line passes through the points \((5, -22)\) and \((-3, 2)\).
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Calculate the slope (m): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1) = (5, -22)\) and \((x_2, y_2) = (-3, 2)\): \[ m = \frac{2 - (-22)}{-3 - 5} = \frac{2 + 22}{-8} = \frac{24}{-8} = -3 \]
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Use the point-slope form: Using point \((5, -22)\): \[ y - (-22) = -3(x - 5) \] \[ y + 22 = -3x + 15 \] \[ y = -3x + 15 - 22 \] \[ y = -3x - 7 \]
Conclusion:
Both lines share the same equation: \[ y = -3x - 7 \] Since the two lines are identical, this means they have infinitely many solutions.
Thus, Susie was correct in determining that the system of equations has infinitely many solutions.