Question
Which of these graphs shows that the linear system −2x+y=6
−
2
𝑥
+
𝑦
=
6
and 4x−2y=−12
4
𝑥
−
2
𝑦
=
−
12
has an infinite number of solutions?(1 point)
Responses
Image with description 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.
−
2
𝑥
+
𝑦
=
6
and 4x−2y=−12
4
𝑥
−
2
𝑦
=
−
12
has an infinite number of solutions?(1 point)
Responses
Image with description 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.
Answers
Answered by
GPT-4o mini
To determine which graph shows that the linear system has an infinite number of solutions, we first need to analyze the given equations:
1. \(-2x + y = 6\)
2. \(4x - 2y = -12\)
We can see if these equations represent the same line, as overlapping lines indicate an infinite number of solutions.
### Step 1: Rearranging to understand the equations
Let's rewrite both equations in slope-intercept form \(y = mx + b\).
For the first equation:
\[
y = 2x + 6 \quad (rearranging)
\]
For the second equation, we can simplify it:
\[
4x - 2y = -12 \quad (divide the entire equation by 2)
\]
\[
2x - y = -6 \quad (rearranging)
\]
Adding \(y\) to both sides:
\[
y = 2x + 6
\]
### Conclusion
Both equations are equivalent: \(y = 2x + 6\). This means that they represent the same line in the coordinate plane.
### Graph analysis:
- The graph that has an infinite number of solutions will show one line on top of the other, meaning they have the same slope and y-intercept.
### Based on your description:
The graph that shows a solid upward slanting line passing through \((-3, 0)\) and \((0, 6)\) (which represents the first equation) and a dotted line that also passes through \((0, 6)\) and \((1, 7)\) does not show that these lines overlap. Thus it indicates that there is no infinite solution.
For there to be an infinite number of solutions, the displayed graph must just show one line (the same line) or one solid line without a separate one. If instead, there were two solid lines on top of each other, that would indicate infinite solutions.
Thus, **the correct graph showing infinite solutions would be one where the two lines are coincident, not intersecting**. The description given does not suggest an infinite number of solutions since there are two separate lines. If you see a graph where only one line is drawn or both lines are exactly the same, that would be your answer.
1. \(-2x + y = 6\)
2. \(4x - 2y = -12\)
We can see if these equations represent the same line, as overlapping lines indicate an infinite number of solutions.
### Step 1: Rearranging to understand the equations
Let's rewrite both equations in slope-intercept form \(y = mx + b\).
For the first equation:
\[
y = 2x + 6 \quad (rearranging)
\]
For the second equation, we can simplify it:
\[
4x - 2y = -12 \quad (divide the entire equation by 2)
\]
\[
2x - y = -6 \quad (rearranging)
\]
Adding \(y\) to both sides:
\[
y = 2x + 6
\]
### Conclusion
Both equations are equivalent: \(y = 2x + 6\). This means that they represent the same line in the coordinate plane.
### Graph analysis:
- The graph that has an infinite number of solutions will show one line on top of the other, meaning they have the same slope and y-intercept.
### Based on your description:
The graph that shows a solid upward slanting line passing through \((-3, 0)\) and \((0, 6)\) (which represents the first equation) and a dotted line that also passes through \((0, 6)\) and \((1, 7)\) does not show that these lines overlap. Thus it indicates that there is no infinite solution.
For there to be an infinite number of solutions, the displayed graph must just show one line (the same line) or one solid line without a separate one. If instead, there were two solid lines on top of each other, that would indicate infinite solutions.
Thus, **the correct graph showing infinite solutions would be one where the two lines are coincident, not intersecting**. The description given does not suggest an infinite number of solutions since there are two separate lines. If you see a graph where only one line is drawn or both lines are exactly the same, that would be your answer.