To determine if the system of equations has infinitely many solutions, we need to check if the two lines are the same (i.e., they have the same slope and y-intercept), which occurs when they are identical.
First, let's calculate the slopes of both lines using the points provided:
For the first line passing through (2, -13) and (-2, -1):
- Calculate the slope \( m_1 \): \[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-13)}{-2 - 2} = \frac{12}{-4} = -3 \]
Now for the second line passing through (5, -22) and (-3, 2):
- Calculate the slope \( m_2 \): \[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - (-22)}{-3 - 5} = \frac{24}{-8} = -3 \]
Both lines have the same slope (\( m_1 = m_2 = -3 \)).
Next, we need to check if they are identical (i.e., they have the same y-intercept). We can find the equations of both lines in slope-intercept form (y = mx + b).
For the first line: Using point (2, -13): \[ y - (-13) = -3(x - 2) \ y + 13 = -3x + 6 \ y = -3x - 7 \]
For the second line: Using point (5, -22): \[ y - (-22) = -3(x - 5) \ y + 22 = -3x + 15 \ y = -3x - 7 \]
Both lines have the equation \( y = -3x - 7 \), which means they are indeed the same line.
Since the two lines are identical, Susie is correct: there are infinitely many solutions to this system of equations.
Answer: Yes, Susie is correct.