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./././././

double check

GPT-4o mini · Answer

Let's review each question one by one based on the provided summaries.

First Question

Which of the following points, if any, are solutions to the system of equations?

Given equations: \(y = 2x + 2\) and \(y = 2x - 4\).
Points to check: (0, 2) and (0, −4).
For (0, 2):
- Substitute \(x = 0\) into both equations:
  - First equation: \(y = 2(0) + 2 = 2\) (this point satisfies the first equation).
  - Second equation: \(y = 2(0) - 4 = -4\) (this point does not satisfy the second equation).
For (0, −4):
- Substitute \(x = 0\) into both equations:
  - First equation: \(y = 2(0) + 2 = 2\) (this point does not satisfy the first equation).
  - Second equation: \(y = 2(0) - 4 = -4\) (this point satisfies the second equation).
Conclusion: Neither of these points are solutions.

Second Question

Which of the following points can be used to best estimate the time when the divers are at the same depth?

Possible points: (3.67, −7), (3, −6), (3.5, −6.5), (3.5, 6.5).

To determine the best answer, ideally one would need to see the graph to identify where the two lines intersect, indicating the point where the divers are at the same depth.

Assuming typical depths and intersections, either (3, −6) or (3.5, −6.5) could be reasonable estimates, as they are both negative depths (assuming the graph indicates depth below 0). Without the graph, it's challenging to determine the exact answer, but:

Best estimate might be (3.5, −6.5), assuming it lies reasonably in the midsection of depth.

Third Question

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

Possible solutions: (3, 7), (3.5, 7.5), (−3, 7), (−3.5, 7.5).

Assuming the intersection of the lines shown in the image is at a point close to (3, 7), it might be reasonable to select:

(3, 7) as a potential solution unless the intersection clearly points to another coordinate.

Fourth Question

Which statement best describes the system of equations {y = -3x + 7; y = -3x - 7}?

The equations have the same slope of \(-3\) but different y-intercepts (7 and -7).
Conclusion: 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.

Fifth Question

Which of the following systems of linear equations has no solution? Choices:

{4x + y = 17; 4x - y = 3}
{4x - y = -17; 4x + y = 3}
{4x - y = 17; 4x + y = 3}
{4x - y = 17; 4x - y = 3}

A system has no solution if the lines are parallel (same slope, different intercepts).
The third choice has the same slope, but different y-intercepts (17 and 3).
So, {4x - y = 17, 4x + y = 3} or possibly others depending on graph verification.

In conclusion:

Neither of these points are solutions.
(3.5, −6.5)
(3, 7)
No solutions, because the lines are parallel.
{4x - y = 17; 4x + y = 3} might represent no solutions depending on precise graphical layout.

If you happen to have further context from the graphs mentioned or need evidenced assessments based on their detailed representations, please let me know!