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
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.
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.
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.
Responses
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.
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.
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.
Answers
Answered by
GPT-4o mini
To determine which graph indicates that the linear system has an infinite number of solutions, we need to understand the meaning of the solutions in a linear system.
1. **Infinite Solutions:** This occurs when the two equations represent the same line. In such a case, every point on that line is a solution to the system.
2. **No Solutions:** This happens when the two lines are parallel but do not intersect.
3. **One Solution:** This occurs when the two lines intersect at a single point.
Now, let's analyze the graphs based on the descriptions provided:
1. **First Graph:** It shows one line only. This does not represent a system of equations.
2. **Second Graph:** It shows two intersecting lines. This indicates that there is only one solution since the lines cross each other.
3. **Third Graph:** It shows two parallel lines. Since parallel lines never intersect, this configuration indicates that there is no solution.
None of the graphs indicate that the system of equations has an infinite number of solutions, which would require two lines to overlap completely (one line represented in two ways).
If one of the graphs represented identical lines, that would show an infinite number of solutions. However, according to the descriptions provided, there doesn’t seem to be a graph representing this scenario.
1. **Infinite Solutions:** This occurs when the two equations represent the same line. In such a case, every point on that line is a solution to the system.
2. **No Solutions:** This happens when the two lines are parallel but do not intersect.
3. **One Solution:** This occurs when the two lines intersect at a single point.
Now, let's analyze the graphs based on the descriptions provided:
1. **First Graph:** It shows one line only. This does not represent a system of equations.
2. **Second Graph:** It shows two intersecting lines. This indicates that there is only one solution since the lines cross each other.
3. **Third Graph:** It shows two parallel lines. Since parallel lines never intersect, this configuration indicates that there is no solution.
None of the graphs indicate that the system of equations has an infinite number of solutions, which would require two lines to overlap completely (one line represented in two ways).
If one of the graphs represented identical lines, that would show an infinite number of solutions. However, according to the descriptions provided, there doesn’t seem to be a graph representing this scenario.
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