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././././.
1 answer
Let's break down each question based on your description, even though I can't visually interpret the graphs.
Which of the following points, if any, are solutions to the system of equations \( y = 2x + 2 \) and \( y = 2x - 4 \)?
To determine if a point is a solution, we can substitute the x-value of each point into both equations to see if they yield the same y-value.
The equations are parallel (since they have the same slope, 2, but different y-intercepts). Therefore, there are no points that solve this system because parallel lines never intersect.
Answer: Neither of these points are solutions.
Which of the following points can be used to best estimate the time when the divers are at the same depth?
To identify when the divers are at the same depth, you need to look for points where the y-values (depth) are equal. Based on common practice with estimation tied to coordinate points, closely observe proximity in y-values.
After considering the options, if there’s an indication or trend in the points presented, choose the one where the y-values are close to representing the same depth.
(3.5, -6.5) would likely be the best choice, as it is a midpoint of some depth represented around that area between known depths (instead of those leading to positive heights or markedly negative outside the defined ranges).
Answer: (3.5, -6.5)
What is the best approximate solution to the system of equations graphed?
Generally, the solution to a system of linear equations occurs at the intersection of the graphs. The choice that is closest to where the lines intersect should be selected.
If you need to decide where the two lines intersect based on the provided coordinates, choose the point that most likely represents an intersection based on the provided options.
If the earlier mention of \( (3, 7) \) looks valid based on visual clues, then choose that one.
Answer: (3, 7)
Which statement best describes the system of equations \( y = -3x + 7 \) and \( y = -3x - 7 \)?
The slopes are identical (-3) but the y-intercepts differ (7 vs. -7). This indicates the lines are parallel, which means there won't be any solutions as they will never meet.
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.
Which of the following systems of linear equations has no solution?
A system has no solution if the equations represent parallel lines. Given two lines with the same slope but differing y-intercepts will yield no solution.
The option that reflects that formation is: \( {4x - y = 17 \quad \text{and} \quad 4x + y = 3} \) does not seem like the right form. However, analyze the others expressed in conflicting slope forms.
The set which would be similar in slope but opposing in direction will be the expected option here: \( {4x - y = 17 \quad \text{and} \quad 4x - y = 3} \). Identify based on conflicting coefficients staying parallel, using the visual cues shared earlier.
Answer: \( {4x - y = 17 \quad \text{and} \quad 4x - y = 3} \)
Final answers summarized:
Neither of these points are solutions.
(3.5, -6.5)
(3, 7)
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.
