Question

Susie determined that the system of equations below have infinite solutions

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

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

Answer 1

To determine if the system of equations has infinite solutions, we can first find the equations of the lines passing through the given points.

For the first line passing through (2, -13) and (-2, 1):
Let's find the slope first:
m = (y2 - y1) / (x2 - x1)
m = (1 - (-13)) / (-2 - 2)
m = 14 / -4
m = -7/2

Now we can use the point-slope form of the equation of a line:
y - y1 = m(x - x1)
y - (-13) = (-7/2)(x - 2)
y + 13 = (-7/2)x + 7
y = (-7/2)x - 6

For the second line passing through the points (5, -22) and (-3, 2):
Let's find the slope first:
m = (y2 - y1) / (x2 - x1)
m = (2 - (-22)) / (-3 - 5)
m = 24 / -8
m = -3

Now we can use the point-slope form of the equation of a line:
y - y1 = m(x - x1)
y - 2 = (-3)(x + 3)
y - 2 = -3x - 9
y = -3x - 7

Now we have the equations of the two lines:
Line 1: y = (-7/2)x - 6
Line 2: y = -3x - 7

Since the slopes of the two lines are different, they are not parallel and will intersect at a single point. Therefore, the system of equations does not have infinite solutions.