Which statement best describes the system of equations {y=−3x+7y=−3x−7

Question

?(1 point)
Responses

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, 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.

Answer 1

To analyze the system of equations given:

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

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

• For the first equation, the slope \( m \) is -3 and the y-intercept \( b \) is 7 (the point (0, 7)).
• For the second equation, the slope \( m \) is also -3, but the y-intercept \( b \) is -7 (the point (0, -7)).

Since both equations have the same slope of -3 but different y-intercepts (7 and -7), they are parallel lines. Parallel lines do not intersect and thus have no solutions.

The correct statement 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 2

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 3

To determine which of the given systems of linear equations has no solution, we need to analyze each set of equations to see if they represent parallel lines (which have the same slope but different y-intercepts).

System 1:

1. \( 4x - y = 17 \)
2. \( 4x + y = 3 \)

Rearranging these in slope-intercept form (y = mx + b):

1. \( y = 4x - 17 \) (slope = 4)
2. \( y = -4x + 3 \) (slope = -4)

These lines intersect; therefore, they have a solution.

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System 2:

1. \( 4x - y = -17 \)
2. \( 4x + y = 3 \)

Rearranging:

1. \( y = 4x + 17 \) (slope = 4)
2. \( y = -4x + 3 \) (slope = -4)

These lines intersect; therefore, they have a solution.

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System 3:

1. \( 4x + y = 17 \)
2. \( 4x - y = 3 \)

Rearranging:

1. \( y = -4x + 17 \) (slope = -4)
2. \( y = 4x - 3 \) (slope = 4)

These lines intersect; therefore, they have a solution.

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System 4:

1. \( 4x - y = 17 \)
2. \( 4x - y = 3 \)

Rearranging:

1. \( y = 4x - 17 \) (slope = 4)
2. \( y = 4x - 3 \) (slope = 4)

Both equations have the same slope of 4 and the same slope, hence they are parallel. These lines do not intersect and thus have no solution.

Conclusion:

The system with no solution is:

\({4x - y = 17, 4x - y = 3}\)

Answer 4

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

(1 point)
Responses

(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)
left parenthesis negative 3.5 comma 7.5 right parenthesis

(3, 7)

Answer 5

To provide the best approximate solution to a system of equations based on a graph, one typically needs to visually analyze the intersection point of the lines represented by the equations. However, since I do not have the graphical representation of the equations in this context, I can guide you on how to determine the best approximate solution if you were looking at the graph.

1. Identify the Intersection Point: Look for the point where the two lines intersect.
2. Estimate the Coordinates: Read the approximate coordinates of the intersection point from the graph. This would typically involve finding the x-coordinate and y-coordinate of the point.

Given the options you provided, you are looking for an intersection point among the following coordinates:

• (3.5, 7.5)
• (−3, 7)
• (−3.5, 7.5)
• (3, 7)

Steps to Approximate:

• If the intersection point is close to (3.5, 7.5), that would be your answer.
• If the intersection is near (−3, 7), that would be your answer.
• Repeat the process with the other options.

Conclusion:

Without the graph to see where the lines intersect, I cannot definitively select which of these options is the best approximate solution. If you can visually determine the point of intersection, that would guide you to the correct response. If you provide more context or details about how the lines appear on the graph, I would be happy to help further!