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.

Answer 1

To answer your questions, I will break down each part based on the details provided.

Question 1: Solutions to the system of equations {y = 2x + 2, y = 2x - 4}

• Point (0, 2): Substitute x = 0 into both equations.

  • For y = 2x + 2: y = 2(0) + 2 = 2 ✓ (solution)
  • For y = 2x - 4: y = 2(0) - 4 = -4 ✗ (not a solution)

• Point (0, -4): Substitute x = 0.

  • For y = 2x + 2: y = 2(0) + 2 = 2 ✗ (not a solution)
  • For y = 2x - 4: y = 2(0) - 4 = -4 ✓ (solution)

Answer: (0, 2) is a solution to one equation, and (0, -4) is a solution to the other equation. Therefore, Neither of these points are solutions.

Question 2: Time when divers are at the same depth

To estimate the time when the divers are at the same depth, you would look for the point where the two lines intersect or get close in value in the y-axis since y represents their depth.

• (3.67, -7): Could be a reasonable estimate depending on the graph.
• (3, -6): Likewise.
• (3.5, -6.5): Centrally located within the potential intersection area.
• (3.5, 6.5): This value doesn’t seem to fit if we’re considering a negative depth for a dive into a pool.

Given these points, (3.5, -6.5) seems the most reasonable estimate.

Question 3: Best approximate solution to the system of equations

Without an image to assess the actual intersection of the two lines, I can only provide guidance based on potential choices.

• (3, 7): Solution candidate.
• (3.5, 7.5): Another possible solution.
• (−3, 7): Unlikely if lines are intersecting positively.
• (−3.5, 7.5): Should be verified against the graph.

Based on the intersections, you would pick the point that looks nearest to where they cross each other.

Question 4: Statement describing the system of equations {y = -3x + 7, y = -3x - 7}

Both equations have a slope of -3, which indicates they are parallel lines (since they have the same slope but different y-intercepts).

Answer: 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.

Question 5: Systems of linear equations with no solution

This question asks you to identify which system has no solutions, i.e., the lines are parallel.

• {4x + y = 17, 4x - y = 3} → Same slope.
• {4x - y = -17, 4x + y = 3} → Different slope.
• {4x - y = 17, 4x + y = 3} → Different slope.
• {4x - y = 17, 4x - y = 3} → Identical equations, infinite solutions.

The first option has equations of the same slope and y-intercepts, indicating parallel lines.

Answer: {4x + y = 17, 4x - y = 3}.

Conclusion:

Based on the analysis, the answers are:

1. Neither of these points are solutions.
2. (3.5, -6.5).
3. The best approximate solution would depend on the graph which is missing.
4. 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. {4x + y = 17, 4x - y = 3}.