Bot GPT 4-o mini To graph the equation \(9 - x = 32y\) using slope-intercept form, we first need to rearrange the equation into the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Starting with the given equation:
\[ 9 - x = 32y \]
We can isolate \(y\) by performing the following steps:
- Subtract 9 from both sides:
\[ -x = 32y - 9 \]
- Add \(x\) to both sides:
\[ 0 = 32y + x - 9 \]
- Rearrange to isolate \(y\):
\[ 32y = -x + 9 \]
- Divide everything by 32 to solve for \(y\):
\[ y = -\frac{1}{32}x + \frac{9}{32} \]
Now we have our equation in slope-intercept form:
\[ y = -\frac{1}{32}x + \frac{9}{32} \]
From this equation, we can determine that:
- The slope \(m = -\frac{1}{32}\) (indicating a very gentle downward slope).
- The y-intercept \(b = \frac{9}{32}\), which is approximately \(0.28125\).
To graph this, you can plot the y-intercept at \((0, \frac{9}{32})\) and then use the slope to find another point. Since the slope is \(-\frac{1}{32}\), you can go down 1 unit on the y-axis and move 32 units to the right on the x-axis to find another point.
Thus, if we start at \( (0, \frac{9}{32}) \), moving right 32 units and down 1 unit leads us to the point \((32, -1)\).
You can also find points for other values of \(x\) or use the points calculated earlier to validate against the graph described in the responses.
If none of the responses fit the correct graph based on these calculations, please recheck the options provided for inconsistencies.
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