Question
hich 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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Answers
GPT-4o mini
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, it is essential to understand that this situation arises when the two equations represent the same line.
### Step 1: Convert the first equation into slope-intercept form
The first equation is \(3x + 2y = 12\).
1. Subtract \(3x\) from both sides:
\[
2y = -3x + 12
\]
2. Divide everything by 2:
\[
y = -\frac{3}{2}x + 6
\]
### Step 2: Compare the equations
Now, we compare the modified version of the first equation \(y = -\frac{3}{2}x + 6\) with the second equation \(y = -\frac{3}{2}x + 6\).
Both equations are identical. Therefore, they represent the same line.
### Conclusion
The graph that shows these two equations has the same slope and y-intercept will depict a single line, confirming that the system has an infinite number of solutions (i.e., every point on the line is a solution).
Look for a graph where both lines overlap completely, indicating they are the same line. That is the graph that shows the system \(3x + 2y = 12\) and \(y = -\frac{3}{2}x + 6\) has an infinite number of solutions.
### Step 1: Convert the first equation into slope-intercept form
The first equation is \(3x + 2y = 12\).
1. Subtract \(3x\) from both sides:
\[
2y = -3x + 12
\]
2. Divide everything by 2:
\[
y = -\frac{3}{2}x + 6
\]
### Step 2: Compare the equations
Now, we compare the modified version of the first equation \(y = -\frac{3}{2}x + 6\) with the second equation \(y = -\frac{3}{2}x + 6\).
Both equations are identical. Therefore, they represent the same line.
### Conclusion
The graph that shows these two equations has the same slope and y-intercept will depict a single line, confirming that the system has an infinite number of solutions (i.e., every point on the line is a solution).
Look for a graph where both lines overlap completely, indicating they are the same line. That is the graph that shows the system \(3x + 2y = 12\) and \(y = -\frac{3}{2}x + 6\) has an infinite number of solutions.