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Question

One Solution, No Solution, or Many Solutions Quick Check
4 of 54 of 5 Items

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. 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.

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.
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Answer 1

To determine which graph shows that the linear system given by the equations \( -2x + y = 6 \) and \( 4x - 2y = -12 \) has an infinite number of solutions, we need to recognize that this occurs when the two lines represented by these equations are actually the same line. This means they have the same slope and y-intercept, making them coincident.

Let's analyze the equations:

1. The first equation can be rearranged into slope-intercept form (y = mx + b): \[ y = 2x + 6 \] The slope is 2 and the y-intercept is 6.

2. The second equation: \[ 4x - 2y = -12 \] can be rearranged as: \[ -2y = -4x - 12 \quad \Rightarrow \quad 2y = 4x + 12 \quad \Rightarrow \quad y = 2x + 6 \] This shows that the second equation simplifies to the same line as the first.

Now, since both equations represent the same line, we can conclude that the system has infinitely many solutions.

The graph that would represent this situation is one where there is only a single line shown, instead of two distinct lines or parallel lines.

Based on your descriptions:

• The first and second options depict intersecting or parallel lines, which would not indicate infinitely many solutions.
• The third option depicts a single line, which could represent both equations, thus having infinitely many solutions.
• The fourth option depicts another distinct single line.

Therefore, the correct graph indicating that the system has an infinite number of solutions would be the third option where only one line is shown, represented in multiple forms but being the same line.