Question

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?

Yes, Susie is correct.
No, there is no point of intersection.
She may or may not be correct as the lines possibly cross.
No, the lines are parallel.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To determine if Susie is correct, we can find the equations of the two lines and see if they are the same or if they intersect at a single point.

Let's find the equation of the first line going through the points (2, -13) and (-2, -1).
Using the point-slope formula, the slope of the line is:

m = (change in y) / (change in x)
m = (-1 - (-13)) / (-2 - 2)
m = 12 / (-4)
m = -3

We can use the point-slope form of a line to find the equation:
y - y1 = m(x - x1), where (x1, y1) is a point on the line.
Using the point (2, -13), the equation of the first line is:
y - (-13) = -3(x - 2)
y + 13 = -3x + 6
y = -3x - 7

Now let's find the equation of the second line going through the points (5, -22) and (-3, 2).
Using the same method, the slope of the line is:

m = (2 - (-22)) / (-3 - 5)
m = 24 / (-8)
m = -3

Using the point (5, -22), the equation of the second line is:
y - (-22) = -3(x - 5)
y + 22 = -3x + 15
y = -3x - 7

Now we can see that the equations of both lines are identical: y = -3x - 7. Therefore, the two lines are the same line, and they intersect at every point on that line. So, Susie is correct. The system of equations has infinitely many solutions.

Answer 2

are you sure?

Answer 3

[AI-generated answer — Bot GPT 3.5]

Apologies for the previous incorrect response. The equation of the first line using the points (2, -13) and (-2, -1) is:

m = (change in y) / (change in x)
m = (-1 - (-13)) / (-2 - 2)
m = 12 / (-4)
m = -3

Using the point-slope form of a line with the point (2, -13), the equation of the first line is:

y - y1 = m(x - x1)
y - (-13) = -3(x - 2)
y + 13 = -3x + 6
y = -3x - 7

The equation of the second line using the points (5, -22) and (-3, 2) is:

m = (change in y) / (change in x)
m = (2 - (-22)) / (-3 - 5)
m = 24 / (-8)
m = -3

Using the point-slope form of a line with the point (5, -22), the equation of the second line is:

y - y1 = m(x - x1)
y - (-22) = -3(x - 5)
y + 22 = -3x + 15
y = -3x - 7

Upon comparing the equations of both lines, we can see that they are identical: y = -3x - 7. Therefore, the two lines are the same line.

Since the two lines are the same line, they intersect at every point on that line. So, Susie is correct. The system of equations has infinitely many solutions.

Answer 4

Put the steps in order that are used to solve the following systems of equations by substitution.

−7x−2y=−13 x−2y=11

Answer 5

[AI-generated answer — Bot GPT 3.5]

Here are the steps in order to solve the system of equations by substitution:

1. Choose one of the equations and solve it for one variable in terms of the other variable. Let's solve the second equation for x:
x - 2y = 11
x = 2y + 11

2. Substitute the expression for the variable found in step 1 into the other equation. Let's substitute x = 2y + 11 into the first equation:
-7x - 2y = -13
-7(2y + 11) - 2y = -13

3. Simplify the equation obtained in step 2 and solve for y. Using the distributive property:
-14y - 77 - 2y = -13
-16y - 77 = -13
-16y = 64
y = -4

4. Substitute the value of y found in step 3 back into one of the original equations to solve for x. Let's use the second equation:
x - 2y = 11
x - 2(-4) = 11
x + 8 = 11
x = 3

5. The solution to the system of equations is the ordered pair (x, y). Therefore, the solution is (3, -4).

The correct order of steps to solve the system of equations by substitution is: 1, 2, 3, 4, 5.

Answer 6

[AI-generated answer — Step-by-Step Bot]

To determine if Susie is correct, we can find the slopes of the two lines and check if they are equal.

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)

Let's calculate the slopes of the two lines:

The first line passes through the points (2, -13) and (-2, -1):
slope1 = (-1 - (-13)) / (-2 - 2)
= (-1 + 13) / (-4)
= 12 / -4
= -3

The second line passes through the points (5, -22) and (-3, 2):
slope2 = (2 - (-22)) / (-3 - 5)
= (2 + 22) / (-8)
= 24 / -8
= -3

Since both slopes are equal (-3), the two lines are parallel. Therefore, there is no point of intersection, and Susie is not correct. The correct answer is: "No, there is no point of intersection."

Answer 7

[AI-generated answer — Explain Bot]

To determine if the system of equations has infinitely many solutions or not, we need to check if the lines are parallel or if they intersect at a unique point.

Let's start by finding the slope of each line. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

For the first line, with points (2, -13) and (-2, -1), the slope is:

m1 = (-1 - (-13)) / (-2 - 2) = 12 / -4 = -3

For the second line, with points (5, -22) and (-3, 2), the slope is:

m2 = (2 - (-22)) / (-3 - 5) = 24 / -8 = -3

Since both slopes are the same (-3), we can conclude that the lines are parallel.

When two lines are parallel, they have the same slope but different y-intercepts. This means they will never intersect and thus have no point in common. Therefore, the system of equations does not have any solutions.

So, the correct answer is: No, there is no point of intersection.