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Which of these graphs shows that the linear system 3x+2y=12 and y=-3/2x+6 has an infinite number of solutions?(1 point)

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which one is it

3 answers

For the linear system to have an infinite number of solutions, the two equations must represent the same line. This occurs when the equations are equivalent, meaning they have the same slope and y-intercept.

Let's analyze the given equations:

  1. First equation: \( 3x + 2y = 12 \)

    • To put this in slope-intercept form (y = mx + b), we can solve for y: \[ 2y = -3x + 12 \ y = -\frac{3}{2}x + 6 \]

  2. Second equation: \( y = -\frac{3}{2}x + 6 \)

Both equations are equivalent, as they both simplify to the exact same line, \( y = -\frac{3}{2}x + 6 \).

To find a graph that shows an infinite number of solutions for this system, look for a graph where both lines coincide (the lines overlap completely).

If you have a set of graphs, you will look for:

  • Two lines that are exactly on top of each other (infinite solutions).
  • No visible intersection points or divergence, as they are the same line.

If you provide those graphs or describe them, I can help you identify the correct one.

Which of these graphs shows that the linear system 3x+2y=12 and y=-3/2x+6 has an infinite number of solutions?(1 point)
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To determine which graph shows that the linear system \(3x + 2y = 12\) and \(y = -\frac{3}{2}x + 6\) has an infinite number of solutions, you should look for the graph where both lines overlap completely, indicating that they represent the same line.

Since I can't see the graphs directly, here’s what to look for:

  1. Overlap: The correct graph should show one line lying exactly on top of the other without any gaps or divergence.
  2. No Intersection: You won't see distinct intersection points since both equations give the same line.
  3. Identical Slopes and Intercepts: You will notice both lines have the same slope, which is \(-\frac{3}{2}\), and the same y-intercept, which is \(6\).

If you can display or describe the graphs, I can help you identify which one meets these criteria. Otherwise, choose the graph where the two lines appear to be the same.