To graph the equation \( 9 - x = \frac{3}{2} y \) using the slope-intercept form, we first need to rearrange it into the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Starting with the given equation:
\[ 9 - x = \frac{3}{2} y \]
We can isolate \( y \) by multiplying both sides by \( \frac{2}{3} \):
\[ \frac{2}{3}(9 - x) = y \]
Distributing \( \frac{2}{3} \):
\[ y = \frac{2}{3} \cdot 9 - \frac{2}{3} x \] \[ y = 6 - \frac{2}{3} x \]
Now, we can write it in slope-intercept form:
\[ y = -\frac{2}{3} x + 6 \]
From this equation, we can identify:
- The slope \( m = -\frac{2}{3} \)
- The y-intercept \( b = 6 \) (meaning the line crosses the y-axis at (0, 6))
Steps to Graph:
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Plot the y-intercept (0, 6).
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Use the slope to find another point. The slope of \( -\frac{2}{3} \) means that for every 3 units you move to the right (positive x-direction), you move down 2 units (negative y-direction).
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Starting from (0, 6):
- Move 3 units to the right: (3, 6)
- Move down 2 units: (3, 4)
- Plot the point (3, 4).
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Draw a line through the points (0, 6) and (3, 4).
Final Graph:
The line you graphed represents the equation \( y = -\frac{2}{3}x + 6 \), demonstrating the relationship defined by the original equation \( 9 - x = \frac{3}{2} y \).
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