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

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.
Both of these points are solutions.

(0,2)

Answer 1

To determine whether the given points \((0, -4)\) and \((0, 2)\) are solutions to the system of equations:

1. Equations:

   • \(y = 2x + 2\)
   • \(y = 2x - 4\)

2. Test Point (0, -4):

   1. Substitute \(x = 0\) into the first equation: \[ y = 2(0) + 2 = 2 \] The output for this equation is \(2\), which does not equal \(-4\). So, \((0, -4)\) is not a solution to the first equation.

   2. Substitute \(x = 0\) into the second equation: \[ y = 2(0) - 4 = -4 \] The output for this equation is \(-4\), which means \((0, -4)\) is a solution to the second equation.

3. Test Point (0, 2):

   1. Substitute \(x = 0\) into the first equation: \[ y = 2(0) + 2 = 2 \] The output for this equation is \(2\), which matches the point \((0, 2)\). So, \((0, 2)\) is a solution to the first equation.

   2. Substitute \(x = 0\) into the second equation: \[ y = 2(0) - 4 = -4 \] The output for this equation is \(-4\), which does not equal \(2\). So, \((0, 2)\) is not a solution to the second equation.

Conclusion:

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

Therefore, the response 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 6.5 right parenthesis

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

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

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

Answer 3

To determine which of the points can be used to best estimate the time when the divers are at the same depth, we need to analyze each point in the context of the scenario:

1. Understanding the scenario:

   • One student dives in 3 minutes before the buzzer, which means their diving time would start at \(t = -3\) and continue afterward.
   • The second student dives in at the moment the buzzer goes off, so their diving time starts at \(t = 0\).

2. Analyzing the points:

   • (3.5, 6.5): This point indicates that 3.5 minutes after the buzzer, they are at a height of 6.5 feet. This could represent the first student being at 6.5 feet above ground, which is unlikely to indicate depth.
   • (3, -6): This point indicates that 3 minutes after the buzzer, they are at a depth of -6 feet. If we assume that depth below the water's surface is negative, this could indicate that the first diver (who dove in 3 minutes before) and is at a depth of 6 feet.
   • (3.67, -7): This point suggests that at 3.67 minutes after the buzzer, they are at a depth of -7 feet. This also could represent a potential point where both divers could be at the same depth.
   • (3.5, -6.5): This indicates that 3.5 minutes after the buzzer, they are at a depth of -6.5 feet. This might be another potential height where they might intersect in depth.

3. Considering the Context: Since we want to find a point where both divers are at the same depth, we need to look at the negative depths, as they both enter the water.

• The points \((3, -6)\) and \((3.67, -7)\) indicate a possible intersection in depth where they could meet under the water's surface.

Conclusion:

Without the graph, the most reasonable choices that hint at a probable moment of intersection (where both divers are at the same depth) are:

• (3, -6): This suggests that at the time 3 minutes after the buzzer, one diver is at -6 feet.
• (3.67, -7): This indicates that 3.67 minutes after the buzzer, they are at a depth of -7 feet.

If choosing one based on likely accurate intersection, (3, -6) seems like a good estimate since it directly correlates with the "moment" right after diving in just as the buzzer goes off. However, if precision in time after the buzzer is a factor, (3.67, -7) might be closer to an estimation of when they meet.

Given that choices often emphasize the more rounded numbers for estimations and understanding, (3, -6) could be the best estimate of the time when the divers are at the same depth based on the context provided.

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)
left parenthesis 3 comma 7 right parenthesis

Answer 5

To determine the best approximate solution to the system of equations based on their graph, we typically look for the point where the two lines (or curves) intersect. If you have a visual reference for the graph, you would identify the intersection point, which is the solution to the system of equations.

Since I don't have access to the graph you're referring to, here’s how you can evaluate the points typically:

1. Check the coordinates:

   • (3.5, 7.5): This point suggests that at \(x = 3.5\), \(y\) is approximately \(7.5\).
   • (−3, 7): This indicates that at \(x = -3\), \(y\) is approximately \(7\).
   • (−3.5, 7.5): This suggests that at \(x = -3.5\), \(y\) is approximately \(7.5\).
   • (3, 7): This indicates that at \(x = 3\), \(y\) is approximately \(7\).

2. Identify the best approximate solution: The "best" point would generally be the one that visually appears to lie on both lines of the system if graphed.

• If two lines cross at a particular coordinate or seem to be very close to one of these answer options, that would be the best approximate solution.

3. Using context: If one or two of the points are closer to where the lines intersect based on the graph than others, that would typically be your answer.

If you were observing the graph and noted their intersection closely matched one of these options, that would be your best choice. Without that visual context, I recommend you look at the graph and see which of these points best aligns with where the two lines meet each other.

If you provide any additional context or details from the graph indicating where the lines intersect, I could help further refine the answer! Otherwise, you'll need to refer back to the graph to determine which point closely corresponds to the lines' intersection.