Ask a New Question
Menu

Which of these graphs shows that the linear system 3x−2y=12

and −6x+4y=−24
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 negative 4 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 4 comma 0 right parenthesis and left parenthesis 0 comma 6 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 4 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 2 comma 3 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 4 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 2 comma 3 right parenthesis.

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 4 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 4 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 parallel lines are plotted on the plane. A solid upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis. A dotted upward slanting line with arrows at both ends passes through the points left parenthesis 6 comma 7 right parenthesis and left parenthesis 8 comma 10 right parenthesis.

1 answer

To determine which graph shows that the linear system \(3x - 2y = 12\) and \(-6x + 4y = -24\) has an infinite number of solutions, we first note that two lines have an infinite number of solutions if they are coincident, meaning the same line is represented in different forms.

Let's first rewrite both equations:

  1. \(3x - 2y = 12\)
  2. \(-6x + 4y = -24\)

Now, we simplify the second equation:

\(-6x + 4y = -24\)

We can divide the entire equation by -2:

\(\Rightarrow 3x - 2y = 12\)

Notice that both equations are identical: \(3x - 2y = 12\). This means the two lines overlap completely in the coordinate system, indicating an infinite number of solutions.

Now, looking at the responses provided:

  • The graph showing one line consistently would indicate that both equations are represented by the same line, leading to an infinite number of solutions.
  • A graph with two lines that overlap would also indicate an infinite number of solutions.

Since we are looking for a representation that shows both equations are coincident, we need to find the response that displays a single line or two overlapping lines.

Responses that show one single line include:

  1. A single upward slanting line (from points (-4, 0) and (0, 6)).
  2. A single upward slanting line (from points (0, -6) and (4, 0)).

Responses that could show coincidence with two lines include:

  • The graph with two parallel lines does NOT show an infinite number of solutions.
  • The graph with two intersecting lines shows only one point of intersection (not an infinite number).

The choice that indicates the infinite number of solutions is graphing only one line, as this corresponds to the two equations being the same.

Therefore, the response that indicates the linear system has an infinite number of solutions is the one showing one solid line. Based on the descriptions, select the first or third option that mentions a single upward slanting line—although select the first one if it accurately represents the line \(3x - 2y = 12\).