The image shows the graph of the system of linear equations {y=2x+2y=2x−4

Question

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

(1 point)
Responses

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

Both of these points are solutions.
Both of these points are solutions.

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

(0,2)

Answer 1

To determine if the points (0, -4) and (0, 2) are solutions to the system of equations, we need to check each point against the two equations:

1. \( y = 2x + 2 \)
2. \( y = 2x - 4 \)

Check (0, -4):

For the first equation:

• Substitute \( x = 0 \): \[ y = 2(0) + 2 = 2 \] So, the point (0, -4) does not satisfy the first equation because \( -4 \neq 2 \).

For the second equation:

• Substitute \( x = 0 \): \[ y = 2(0) - 4 = -4 \] This point satisfies the second equation.

Check (0, 2):

For the first equation:

• Substitute \( x = 0 \): \[ y = 2(0) + 2 = 2 \] So, the point (0, 2) satisfies the first equation.

For the second equation:

• Substitute \( x = 0 \): \[ y = 2(0) - 4 = -4 \] This point does not satisfy the second equation because \( 2 \neq -4 \).

Conclusion:

• The point (0, -4) is a solution for the second equation but not for the first.
• The point (0, 2) is a solution for the first equation but not for the second.

Therefore, the answer is: Neither of these points are solutions.

Answer 2

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

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

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

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

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

Answer 3

To determine which point can best estimate the time when the divers are at the same depth, we need to analyze the given information about their diving paths:

• The first student dives from the edge of the pool 3 minutes before the buzzer. Assuming the pool edge is at \(0\) height, when time \(x = 0\) (the moment the buzzer sounds), the height would be \(y = 0\) for that diver. However, they would have started their dive \(3\) minutes earlier, so by \(x = -3\), their height would likely be \(0\) as they are diving into the water.

• The second student dives from a board 4 feet above the ground at the moment the buzzer sounds, meaning when \(x = 0\), their height \(y\) is \(4\).

Analyzing Points

Let's briefly look at the given points:

1. (3.5, -6.5): At \(x = 3.5\), \(y = -6.5\). This indicates a depth of 6.5 feet below the water surface.
2. (3.5, 6.5): At \(x = 3.5\), \(y = 6.5\). This would indicate a height above the water, so it’s irrelevant for their same depth.
3. (3.67, -7): At \(x = 3.67\), \(y = -7\). This indicates a depth of 7 feet below the water surface.
4. (3, -6): At \(x = 3\), \(y = -6\). This indicates a depth of 6 feet below the water surface.

Determining the Best Estimate

Since the first student starts at the edge and presumably was at the surface level at the moment the buzzer went off and is moving down, while the second student starts at 4 feet above, they must be moving downward from that initial height.

To find the time when both divers are at the same depth (where their y-values converge), we look for points where both values could represent depths below the water.

Conclusion

Since all the potential points indicate negative heights (depths) for at least one diver, (3, -6) could indicate a reasonable time for both to be at the same depth shortly after the buzzer sound, as it is right at a point in time where the first diver's descent and the second diver's dive would likely meet.

Thus, the best estimate for the time when the divers are at the same depth would be:

(3, −6).

Answer 4

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 .

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.

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 .
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Answer 5

To analyze the given system of equations:

1. \( y = -3x + 7 \)
2. \( y = -3x - 7 \)

Step 1: Identify the slopes and y-intercepts

Both equations are in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• For the first equation \( y = -3x + 7 \):

  • Slope (m): \(-3\)
  • Y-intercept (b): \(7\)

• For the second equation \( y = -3x - 7 \):

  • Slope (m): \(-3\)
  • Y-intercept (b): \(-7\)

Step 2: Analyze the relationship between the two equations

• The slopes of both equations are equal (\(-3\)), indicating that the lines are parallel.
• The y-intercepts are different (\(7\) for the first and \(-7\) for the second).

Conclusion

Since both equations have the same slope and different y-intercepts, the lines are parallel and will never intersect. Therefore, the system of equations has no solutions.

The statement that best describes the system 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."

Answer 6

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

Answer 7

To determine which system of linear equations has no solution, we need to identify systems where the lines represented by the equations are parallel. This occurs when the coefficients of \(x\) and \(y\) are proportional, but the constants are not.

Here are the provided systems analyzed:

1. System 1: \[ \begin{cases} 4x - y = 17 \ 4x - y = 3 \end{cases} \]

   Both equations have the same slope \((4, -1)\) but different y-intercepts (17 and 3), so these lines are parallel and have no solution.

2. System 2: \[ \begin{cases} 4x - y = 17 \ 4x + y = 3 \end{cases} \] The first equation has a slope of 4 (written as \(y = 4x - 17\)) and the second has a slope of -4 (written as \(y = -4x + 3\)). Since these slopes are different, this system has a unique solution.

3. System 3: \[ \begin{cases} 4x - y = -17 \ 4x + y = 3 \end{cases} \] Similarly as above, the first equation has a slope of 4 and the second has a slope of -4. These slopes are also different, indicating a unique solution.

4. System 4: \[ \begin{cases} 4x + y = 17 \ 4x - y = 3 \end{cases} \] In this case, the first equation has a slope of -4 (as \( y = -4x + 17 \)) and the second has a slope of 4 (as \( y = 4x - 3 \)). Again, these slopes are different, so this system also has a unique solution.

Conclusion

The system of equations that has no solution is:

{4x−y=17, 4x−y=3}.