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One Solution, No Solution, or Many Solutions Quick Check

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Question
Which of these graphs shows that the linear system −2x+y=6 and 4x−2y=−12 has an infinite number of solutions?(1 point)
Responses

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. An upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 3 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. An upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 3 comma 0 right parenthesis.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 1 comma 7 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 1 comma 7 right parenthesis.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A solid upward slanting line with arrows at both ends, parallel to the dotted line, passes through origin.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A solid upward slanting line with arrows at both ends, parallel to the dotted line, passes through origin.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. An upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. An upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis.
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1 answer

To determine which graph shows that the linear system has an infinite number of solutions, we need to understand the concept of linear equations and their graphs:

  1. One Solution: This occurs when the two lines intersect at one point.
  2. No Solution: This occurs when the lines are parallel and never intersect.
  3. Infinite Solutions: This occurs when the two lines are the same line (i.e., they overlap completely).

The linear equations given are:

  • \( -2x + y = 6 \)
  • \( 4x - 2y = -12 \)

First, we can manipulate the second equation to see if it simplifies to the first equation:

  1. Rearranging \( 4x - 2y = -12 \): \[ -2y = -4x - 12 \] \[ y = 2x + 6 \]

Now, let's rewrite the first equation \( -2x + y = 6 \) in the form of \( y \): \[ y = 2x + 6 \]

Since both lines can be represented by the same equation \( y = 2x + 6 \), this means they are the same line, indicating that the linear system has an infinite number of solutions.

Now, based on your options:

  • The third option describes two parallel lines, which indicates no solution.
  • The second option has two intersecting lines, indicating one solution.
  • The first option and the fourth option show only single lines.

Thus, the correct response that shows the linear system has an infinite number of solutions should be represented by overlapping lines, but none of the options explicitly state this.

However, if you have to decide based on which line representation suggests the possibility for infinite solutions, you'd be looking for parallel lines that simplify to the same line. Among the options given, you want something that represents two lines that are the same.

If you have access to the images, choose the graph that appears to have two identical lines, as that would indicate an infinite number of solutions, which seems to be missing according to your description. If the second option (if it represents two identical lines instead of intersecting) was the intention, it would then be the answer.

Given the descriptions, the correct graph would be:

  • Two identical lines (not intersecting, but representing the same line).

None of the listed responses seem to accurately indicate this solution, but if a response suggested overlapping lines this would be the desired outcome. If possible, please confirm the images' visual information for accurate identification.