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 with the original equation:
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Rearrange the equation to isolate \(y\): \[ 9 - x = \frac{3}{2}y \] Adding \(x\) to both sides gives: \[ 9 = \frac{3}{2}y + x \] Subtracting \(x\) from both sides gives: \[ 9 - x = \frac{3}{2}y \]
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Now, multiply both sides by \(\frac{2}{3}\) to solve for \(y\): \[ y = \frac{2}{3}(9 - x) \]
Distributing gives: \[ y = \frac{18}{3} - \frac{2}{3}x \] Simplifying: \[ y = 6 - \frac{2}{3}x \]
Now we have the equation in slope-intercept form: \[ y = -\frac{2}{3}x + 6 \] From this equation, we know the slope \(m = -\frac{2}{3}\) and the y-intercept \(b = 6\).
Finding Points for the Graph:
- The y-intercept is \(6\) (point \( (0, 6) \)).
- To find another point, you can choose any value of \(x\). For example, if \(x = 3\): \[ y = 6 - \frac{2}{3}(3) = 6 - 2 = 4 \quad \text{(point \( (3, 4) \))} \]
- Another example, if \(x = 6\): \[ y = 6 - \frac{2}{3}(6) = 6 - 4 = 2 \quad \text{(point \( (6, 2) \))} \]
Plotting the Points:
- The points \( (0, 6) \) and \( (3, 4) \) can be plotted on the coordinate plane.
- Draw a line through these points, extending it in both directions.
Choose from the options given based on the plotted points and line:
- Verify the plotted points and the line characteristics. The correct option should have a line that matches the slope and y-intercept calculated.
Based on the information given, it seems the correct response will describe a line that goes through points like \( (0, 6) \) and \( (3, 4) \) or similar points consistent with a downward slope.
1 answer