Question
Which of these graphs shows that the linear system has an infinite number of solutions?
3x+2y=12
6x+4y=24
(1 point)
3x+2y=12
6x+4y=24
(1 point)
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. **First equation:** \( 3x + 2y = 12 \)
2. **Second equation:** \( 6x + 4y = 24 \)
We can observe that the second equation is actually a multiple of the first. If we multiply the first equation by 2, we get:
\[
2(3x + 2y) = 2(12)
\]
\[
6x + 4y = 24
\]
Since the second equation is simply a scaled version of the first, both equations represent the same line. This means that every point on this line is a solution to both equations, which implies that there are infinitely many solutions.
To visually identify the graphs, the one representing this system will show the two lines overlapping completely or as the same line.
In conclusion, look for a graph where both lines overlap perfectly, indicating that the system has an infinite number of solutions.
1. **First equation:** \( 3x + 2y = 12 \)
2. **Second equation:** \( 6x + 4y = 24 \)
We can observe that the second equation is actually a multiple of the first. If we multiply the first equation by 2, we get:
\[
2(3x + 2y) = 2(12)
\]
\[
6x + 4y = 24
\]
Since the second equation is simply a scaled version of the first, both equations represent the same line. This means that every point on this line is a solution to both equations, which implies that there are infinitely many solutions.
To visually identify the graphs, the one representing this system will show the two lines overlapping completely or as the same line.
In conclusion, look for a graph where both lines overlap perfectly, indicating that the system has an infinite number of solutions.