Use the image to answer the question.

Question

Use the image to answer the question.

A coordinate graph goes from negative 10 to 10 in 1-unit increments on both the x- and y-axes. Two linear equations are graphed, along with two points.

The image shows the graph of the system of linear equations {y=2x+2y=2x−4
. Which of the following points, if any, are solutions to the system of equations?

(1 point)
Responses

(0,2)
open paren 0 comma 2 close paren

Neither of these points are solutions.
Neither of these points are solutions.

(0,−4)
open paren 0 comma negative 4 close paren

Both of these points are solutions.

Use the image to answer the question.

Two intersecting lines are graphed across a coordinate plane.

The graph models the swimming paths of two students, where x represents the time after a buzzer sounds and y represents height from the pool edge. One student dived into the pool from the edge of the pool 3 minutes before the buzzer went off. The second student dived into the pool from a board 4 feet above the ground right when the buzzer went off. Which of the following points can be used to best estimate the time when the divers are at the same depth?

(1 point)
Responses

(3.67, −7)
left parenthesis 3.67 comma negative 7 right parenthesis

(3, −6)
left parenthesis 3 comma negative 6 right parenthesis

(3.5, −6.5)
left parenthesis 3.5 comma negative 6.5 right parenthesis

(3.5, 6.5)
Use the image to answer the question.

Two intersecting lines are graphed across a coordinate plane.

What is the best approximate solution to the system of equations graphed?

(1 point)
Responses

(3, 7)
left parenthesis 3 comma 7 right parenthesis

(3.5, 7.5)
left parenthesis 3.5 comma 7.5 right parenthesis

(−3, 7)
left parenthesis negative 3 comma 7 right parenthesis

(−3.5, 7.5)

Which statement best describes the system of equations {y=−3x+7y=−3x−7
?(1 point)
Responses

Both equations have a slope of −3, but they do not share the same y-intercept. Thus, the system has no solutions because the lines are parallel.
Both equations have a slope of negative 3 , but they do not share the same y -intercept. Thus, the system has no solutions because the lines are parallel.

The equations have different slopes and different y-intercepts. Thus, the system has one solution at (−3,7).
The equations have different slopes and different y -intercepts. Thus, the system has one solution at left parenthesis negative 3 comma 7 right parenthesis .

The equations have different slopes, but they share the same y-intercept, (0,−3). Thus, the system has one solution at (0,−3).
The equations have different slopes, but they share the same y -intercept, left parenthesis 0 comma negative 3 right parenthesis . Thus, the system has one solution at left parenthesis 0 comma negative 3 right parenthesis .

Both equations have a slope of −3, and they share the same y-intercept. Thus, the system has an infinite number of solutions because they are identical lines.
Both equations have a slope of negative 3 , and they share the same y -intercept. Thus, the system has an infinite number of solutions because they are identical lines.

Which of the following systems of linear equations has no solution?(1 point)
Responses

{4x+y=174x−y=3

{4x−y=−174x+y=3

{4x−y=174x+y=3

{4x−y=174x−y=3

Which statement best describes the system of equations {y=−3x+7y=−3x−7
?(1 point)
Responses

Both equations have a slope of −3, but they do not share the same y-intercept. Thus, the system has no solutions because the lines are parallel.
Both equations have a slope of negative 3 , but they do not share the same y -intercept. Thus, the system has no solutions because the lines are parallel.

The equations have different slopes and different y-intercepts. Thus, the system has one solution at (−3,7).
The equations have different slopes and different y -intercepts. Thus, the system has one solution at left parenthesis negative 3 comma 7 right parenthesis .

The equations have different slopes, but they share the same y-intercept, (0,−3). Thus, the system has one solution at (0,−3).
The equations have different slopes, but they share the same y -intercept, left parenthesis 0 comma negative 3 right parenthesis . Thus, the system has one solution at left parenthesis 0 comma negative 3 right parenthesis .

Both equations have a slope of −3, and they share the same y-intercept. Thus, the system has an infinite number of solutions because they are identical lines. ...........

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1 answer
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Answer 1

Bot GPT 4-o mini answered
27 days ago

Here are the answers to your questions:

1. **Which of the following points, if any, are solutions to the system of equations?**
- To see if the points (0, 2) or (0, -4) are solutions to the system of equations \(y = 2x + 2\) or \(y = 2x - 4\):
- For (0, 2):
- In the first equation \(y = 2x + 2\): \(2 = 2(0) + 2 \rightarrow 2 = 2\) (True)
- In the second equation \(y = 2x - 4\): \(2 = 2(0) - 4 \rightarrow 2 = -4\) (False)
- For (0, -4):
- In the first equation \(y = 2x + 2\): \(-4 = 2(0) + 2 \rightarrow -4 = 2\) (False)
- In the second equation \(y = 2x - 4\): \(-4 = 2(0) - 4 \rightarrow -4 = -4\) (True)
- **Correct Response:** Neither of these points are solutions.

2. **Which of the following points can be used to best estimate the time when the divers are at the same depth?**
- To find the point where the divers are at the same depth, you would look for points where both lines intersect. If the graphs, not provided here, show which points are best approximations of the intersection, then:
- Assuming we look for equality in depth (y-values), the best option among the provided choices has to be closest to the point of intersection.
- **Without the graph, I can't give a definitive answer**, but analyze each choice based on proximity to the intersection point.

3. **What is the best approximate solution to the system of equations graphed?**
- **Without the graph**, I can't identify the exact intersection point, but it would generally be the point where the two lines cross, given in the choices.
- Look for the point that visually appears to intersect both lines.

4. **Which statement best describes the system of equations {y=−3x+7, y=−3x−7}?**
- Both equations have the same slope of -3, but different y-intercepts (7 and -7).
- **Correct Response:** Both equations have a slope of −3, but they do not share the same y-intercept. Thus, the system has no solutions because the lines are parallel.

5. **Which of the following systems of linear equations has no solution?**
- Systems of equations have no solution when the equations are parallel (same slopes, different y-intercepts):
- System A: {4x + y = 17, 4x - y = 3} (Different slopes)
- System B: {4x - y = -17, 4x + y = 3} (Different slopes)
- System C: {4x - y = 17, 4x + y = 3} (Different slopes)
- System D: {4x - y = 17, 4x - y = 3} (Same slopes, same y-intercept, thus infinitely many solutions)
- **Correct Response:** {4x - y = 17, 4x + y = 3} or {4x + y = 17, 4x - y = 3}, depending on specific slopes.

Would you like any further clarifications or a deeper explanation?