Susie determined that the system of equations below has infinitely many solutions.
The first line goes through the points (2, -13) and (-2, -1).
The second line goes through the points (5, -22) and (-3, 2).
Is she correct? *
A. No, the lines are parallel.
B. No, there is no point of intersection.
C. She may or may not be correct as the lines possibly cross.
D. Yes, Susie is correct.
To determine if the lines have infinitely many solutions, we need to see if they are the same line.
First, let's find the slope of the first line:
m = (change in y)/(change in x) = (-1 - (-13))/(-2 - 2) = 12/4 = 3
Now, let's find the equation of the first line using the point-slope form:
(y - y1) = m(x - x1)
(y - (-13)) = 3(x - 2)
(y + 13) = 3x - 6
y = 3x - 19
Next, let's find the slope of the second line:
m = (change in y)/(change in x) = (2 - (-22))/(-3 - 5) = 24/-8 = -3
Now, let's find the equation of the second line using the point-slope form:
(y - y1) = m(x - x1)
(y - 2) = -3(x + 3)
y - 2 = -3x - 9
y = -3x - 7
As we can see, the equations of the two lines are different. Therefore, the lines are not the same line and do not have infinitely many solutions.
The correct answer is: B. No, there is no point of intersection.
To determine if Susie is correct, we can find the equations of both lines and check if they intersect at a single point or are parallel.
First, let's find the equation of the first line using the two given points (2, -13) and (-2, -1):
Slope (m) = (y2 - y1) / (x2 - x1)
= (-1 - (-13)) / (-2 - 2)
= 12 / -4
= -3
Using the point-slope form of a linear equation, we have:
y - y1 = m(x - x1)
y - (-13) = -3(x - 2)
y + 13 = -3x + 6
y = -3x - 7 ----(1)
Now, let's find the equation of the second line using the two given points (5, -22) and (-3, 2):
Slope (m) = (y2 - y1) / (x2 - x1)
= (2 - (-22)) / (-3 - 5)
= 24 / -8
= -3
Using the point-slope form of a linear equation, we have:
y - y1 = m(x - x1)
y - 2 = -3(x - (-3))
y - 2 = -3x - 9
y = -3x - 7 ----(2)
Comparing equations (1) and (2), we can see that they have the same slope (-3) and the same y-intercept (-7). Therefore, the two lines are the same line and will intersect at infinitely many points.
So, the correct answer is: D. Yes, Susie is correct.
To determine whether Susie is correct, we need to check if the lines intersect at a single point or if they are parallel.
To find the equation of a line, we can use the formula: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
For the first line that goes through the points (2, -13) and (-2, -1), we can find the slope using the formula: m = (y2 - y1) / (x2 - x1):
m = (-1 - (-13)) / (-2 - 2) = 12 / (-4) = -3.
Using the point-slope form with the point (2, -13):
y - (-13) = -3(x - 2)
y + 13 = -3x + 6
y = -3x -7
So, the equation of the first line is y = -3x - 7.
For the second line that goes through the points (5, -22) and (-3, 2), we can find the slope using the same formula:
m = (2 - (-22)) / (-3 - 5) = 24 / (-8) = -3.
Using the point-slope form with the point (5, -22):
y - (-22) = -3(x - 5)
y + 22 = -3x + 15
y = -3x - 7
So, the equation of the second line is also y = -3x - 7.
Since both lines have the same equation, y = -3x - 7, they are the same line and therefore intersect at infinitely many points.
Therefore, Susie is correct, and the answer is D. Yes, Susie is correct.