Use the image to answer the question.

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.

HELPPPPP

Answer 1

Let's go through the questions based on the information you've provided.

First Question

What points are solutions to the system of equations \(y=2x+2\) and \(y=2x-4\)?

1. (0, 2):

   • Substitute \(x = 0\) into \(y=2x+2\):
     • \(y = 2(0) + 2 = 2\) (valid point)
   • Substitute \(x = 0\) into \(y=2x-4\):
     • \(y = 2(0) - 4 = -4\) (not valid)

   So, (0,2) is not a solution to the system.

2. (0, -4):

   • Substitute \(x = 0\) into \(y=2x+2\):
     • \(y = 2(0) + 2 = 2\) (not valid)
   • Substitute \(x = 0\) into \(y=2x-4\):
     • \(y = 2(0) - 4 = -4\) (valid point)

   So, (0, -4) is not a solution to the system either.

Conclusion: The answer is Neither of these points are solutions.

Second Question

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

• Without the actual image, we can only go on reasoning: A point where \(y\) is negative indicates a depth below the water surface.

Given the choices:

1. (3.67, -7)
2. (3, -6)
3. (3.5, -6.5)
4. (3.5, 6.5) (not likely, as depth would be negative)

Based on this, (3.5, -6.5) might be the best estimate if it seems reasonable between the other values.

Third Question

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

• Without the image, it’s difficult to determine precisely, but you would typically look for the intersection point of the two lines:

1. (3, 7)
2. (3.5, 7.5)
3. (-3, 7)
4. (-3.5, 7.5)

If you can identify the point where the lines intersect on the graph, select that option. In general, from the context, look for a consistent coordinate.

Fourth Question

Which statement describes the system of equations \(y=-3x+7\) and \(y=-3x-7\)?

The slopes of both equations are the same (-3) but have different y-intercepts. Therefore, they are parallel and will not intersect.

The correct answer is: 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?

We look for a pair of equations that are parallel (same slope, different intercepts):

1. {4x+y=17 and 4x−y=3 (Different slopes).
2. {4x−y=−17 and 4x+y=3 (Different slopes).
3. {4x−y=17 and 4x+y=3 (Different slopes).
4. {4x−y=17 and 4x−y=3 (Same slope, same condition; invalid).

The answer is something like: {4x−y=17 and 4x+y=3}.

Feel free to revise with specifics from the graphs or the equations, and I can help narrow it down further!