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Susie determined that the system of equations below has infinitely many solutions. *Use your graphing calculator!

The first line goes through the points (2, -13) and (-2, -1)

The second line goes through the points (5, -22) and (-3, 2)

(1 point)
Responses

Yes, Susie is correct.
Yes, Susie is correct.

No, there is one point of intersection
No, there is one point of intersection

No, the lines are parallel
No, the lines are parallel

She may or may not be correct as the lines possibly cross
She may or may not be correct as the lines possibly cross

1 answer

To determine if Susie's conclusion about the system of equations having infinitely many solutions is correct, we need to analyze the lines defined by the given points.

Finding the equations of the lines:

  1. First Line (through points (2, -13) and (-2, -1)):

    Slope (m) is calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-13)}{-2 - 2} = \frac{12}{-4} = -3 \]

    Using point-slope form (y - y1 = m(x - x1)) with point (2, -13): \[ y + 13 = -3(x - 2) \ y + 13 = -3x + 6 \ y = -3x - 7 \] Thus, the equation of the first line is: \[ y = -3x - 7 \]

  2. Second Line (through points (5, -22) and (-3, 2)):

    Slope (m) is: \[ m = \frac{2 - (-22)}{-3 - 5} = \frac{24}{-8} = -3 \]

    Using point-slope form with point (5, -22): \[ y + 22 = -3(x - 5) \ y + 22 = -3x + 15 \ y = -3x - 7 \] Thus, the equation of the second line is: \[ y = -3x - 7 \]

Conclusion:

Both lines have the same slope (-3) and the same y-intercept (-7), which means they are the same line.

Since both lines have the same equation, they have infinitely many solutions because every point on one line is also a point on the other line.

Therefore, the correct response to Susie's conclusion is:

Yes, Susie is correct.