Question
Which graph shows the linear system of 3x+2y=12 and 6x+4y=24
Answers
GPT-4o mini
To analyze the linear system represented by the equations \(3x + 2y = 12\) and \(6x + 4y = 24\), let's first notice that the second equation, \(6x + 4y = 24\), is simply a multiple of the first equation. Specifically, if we multiply the first equation by 2, we get:
\[
2(3x + 2y) = 2(12) \implies 6x + 4y = 24.
\]
This means both equations represent the same line in the coordinate plane.
Thus, if you were to plot these equations on a graph, you would find that both lines overlap completely, showing that they are the same line. In terms of solutions, this means that there are infinitely many solutions that satisfy both equations, as every point on that line is a solution.
If you are looking at specific graph options, you should identify the graph that shows a single line, not two intersecting lines, indicating that they are the same line.
\[
2(3x + 2y) = 2(12) \implies 6x + 4y = 24.
\]
This means both equations represent the same line in the coordinate plane.
Thus, if you were to plot these equations on a graph, you would find that both lines overlap completely, showing that they are the same line. In terms of solutions, this means that there are infinitely many solutions that satisfy both equations, as every point on that line is a solution.
If you are looking at specific graph options, you should identify the graph that shows a single line, not two intersecting lines, indicating that they are the same line.