Question

Susie determined that the system of equations below has infinity many solutions.
The first like goes through the points (2,-13) and (-2,-1).
The second like goes through the points (5,-22) and (-3,2).
Is she correct?

Answer 1

To determine if Susie is correct, we need to check if the two lines have the same slope and y-intercept.

First, let's find the slope of the first line using the given points (2, -13) and (-2, -1). The slope formula is given by:
m = (y2 - y1) / (x2 - x1)

m = (-1 - (-13)) / (-2 - 2)
m = (-1 + 13) / (-2 - 2)
m = 12 / (-4)
m = -3

Now, let's find the slope of the second line using the given points (5, -22) and (-3, 2).
m = (2 - (-22)) / (-3 - 5)
m = (2 + 22) / (-3 - 5)
m = 24 / (-8)
m = -3

Both lines have a slope of -3, so they have the same slope.

Next, let's find the y-intercept of the first line. We can use the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept. We'll use the point (2, -13) and solve for b:
-13 = -3(2) + b
-13 = -6 + b
b = -13 + 6
b = -7

Now, let's find the y-intercept of the second line. We can use the slope-intercept form again, this time using the point (5, -22):
-22 = -3(5) + b
-22 = -15 + b
b = -22 + 15
b = -7

Both lines have the same y-intercept of -7.

Since the two lines have the same slope (-3) and y-intercept (-7), they are the same line. Therefore, the system of equations has infinitely many solutions. Susie is correct.