Question
whcih of these graphs shouw that the linear system -2x + y = 6 and 4x - 2y = -12 has an infinite number of solutions
A, 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.
B. 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.
C. 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.
D. 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, 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.
B. 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.
C. 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.
D. 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 of the given graphs shows that the linear system \(-2x + y = 6\) and \(4x - 2y = -12\) has an infinite number of solutions, we need to identify whether the two equations represent the same line.
First, let's manipulate the equations to see if they are equivalent:
1. **Rearranging the first equation**:
\[
-2x + y = 6 \implies y = 2x + 6
\]
2. **Now, we manipulate the second equation**:
\[
4x - 2y = -12
\]
Divide the entire equation by 2:
\[
2x - y = -6 \implies y = 2x + 6
\]
Both equations simplify to \(y = 2x + 6\), which shows that they represent the same line. This means the system has an infinite number of solutions, as every point on this line is a solution.
Now let's analyze the options:
- **Option A** shows one solid line and one dotted line that intersect. This indicates that there is a single unique solution (the intersection point), so this does not represent infinite solutions.
- **Option B** shows a single upward slanting line, which indicates only one line, hence one solution (or infinite points that lie on the same line, but not in relation to another line).
- **Option C** also shows a single line, so again only one line is present — therefore this option cannot represent an infinite number of solutions.
- **Option D** shows two parallel lines (one solid and one dotted). Since the lines are parallel and do not intersect, they imply that there are no solutions. However, in this case, we need to ensure they are equivalent to confirm infinite solutions. If both lines are actually the same (i.e., the same slope and y-intercept represented differently), it may create confusion about its terminology.
In summary, to have infinite solutions for a system, the two equations must represent the same line, and since both equations reduce to the same line, the expected graph should depict that.
Since **none of the provided options clearly show two identical lines**, it indicates no option captures the infinite solutions in a clear way. However, if we interpret option D on the premise that both lines are represented as parallel but identical treating one as not identical, it would hint at acknowledgement for equivalency.
Thus, if we remained vigilant to their equivalency and alignment, **option D** could reflect infinite solutions under such a false presentation.
Therefore, the graph that can imply infinite solutions on the basis of representation of being duplicated lines (in parallel misleadingly), we could tentatively say:
**The answer is ~ D.**
First, let's manipulate the equations to see if they are equivalent:
1. **Rearranging the first equation**:
\[
-2x + y = 6 \implies y = 2x + 6
\]
2. **Now, we manipulate the second equation**:
\[
4x - 2y = -12
\]
Divide the entire equation by 2:
\[
2x - y = -6 \implies y = 2x + 6
\]
Both equations simplify to \(y = 2x + 6\), which shows that they represent the same line. This means the system has an infinite number of solutions, as every point on this line is a solution.
Now let's analyze the options:
- **Option A** shows one solid line and one dotted line that intersect. This indicates that there is a single unique solution (the intersection point), so this does not represent infinite solutions.
- **Option B** shows a single upward slanting line, which indicates only one line, hence one solution (or infinite points that lie on the same line, but not in relation to another line).
- **Option C** also shows a single line, so again only one line is present — therefore this option cannot represent an infinite number of solutions.
- **Option D** shows two parallel lines (one solid and one dotted). Since the lines are parallel and do not intersect, they imply that there are no solutions. However, in this case, we need to ensure they are equivalent to confirm infinite solutions. If both lines are actually the same (i.e., the same slope and y-intercept represented differently), it may create confusion about its terminology.
In summary, to have infinite solutions for a system, the two equations must represent the same line, and since both equations reduce to the same line, the expected graph should depict that.
Since **none of the provided options clearly show two identical lines**, it indicates no option captures the infinite solutions in a clear way. However, if we interpret option D on the premise that both lines are represented as parallel but identical treating one as not identical, it would hint at acknowledgement for equivalency.
Thus, if we remained vigilant to their equivalency and alignment, **option D** could reflect infinite solutions under such a false presentation.
Therefore, the graph that can imply infinite solutions on the basis of representation of being duplicated lines (in parallel misleadingly), we could tentatively say:
**The answer is ~ D.**
There are no human answers yet. A form for humans to post answers is coming very soon!