To graph the equation \(9 - x = \frac{3}{2}y\), we first need to convert it into slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Starting from the original equation:
\[ 9 - x = \frac{3}{2}y \]
We can rearrange it to isolate \(y\):
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Move \(9\) and the \(-x\) to the right side: \[ \frac{3}{2}y = 9 - x \]
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Multiply both sides by \(\frac{2}{3}\) to solve for \(y\): \[ y = \frac{2}{3}(9 - x) \] \[ y = \frac{2}{3} \cdot 9 - \frac{2}{3} \cdot x \] \[ y = 6 - \frac{2}{3}x \]
Now we have the slope-intercept form: \[ y = -\frac{2}{3}x + 6 \]
From this equation, we can identify:
- The y-intercept \(b\) is \(6\) (the point where the line crosses the y-axis).
- The slope \(m\) is \(-\frac{2}{3}\), meaning for every 3 units you move to the right, the line goes down by 2 units.
To plot this equation:
- Start at the y-intercept \( (0, 6) \) on the y-axis.
- Using the slope \(-\frac{2}{3}\), from the point \( (0, 6) \):
- Move 3 units to the right (to \(x = 3\)) and 2 units down (to \(y = 4\)), giving the point \( (3, 4) \).
- Draw the line through these two points extending in both directions.
Thus, you can verify the points:
- At \(x = 0, y = 6\) (point on the line).
- At \(x = 3, y = 4\) (another point on the line).
You can also find additional points if needed, but since you are asked for the graph, these points are sufficient to determine the line's positioning. The graph would have a negative slope, confirming the downward slant of the line as noted in your options.
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