Question

Graph the equation 9−x=3/2y using slope-intercept form.(1 point)


An illustration shows a coordinate plane with four quadrants. The x and y axes range from negative 10 to 10 in one unit increments. A downward slanting line is graphed on the coordinate plane. The downward slanting line passes through closed points plotted at left parenthesis 0 comma 6 right parenthesis and left parenthesis 3 comma 4 right parenthesis.
Image with alt text: An illustration shows a coordinate plane with four quadrants. The x and y axes range from negative 10 to 10 in one unit increments. A downward slanting line is graphed on the coordinate plane. The downward slanting line passes through closed points plotted at left parenthesis 0 comma 6 right parenthesis and left parenthesis 3 comma 4 right parenthesis.

A coordinate plane shows the x-axis ranging from negative 9 to 9 in increments of 1 and the y-axis ranging from 11 to negative 11 in increments of 1. A line with arrows at both ends joins two plotted points. The coordinates of the plotted points are as follows: left parenthesis 0 comma 3 right parenthesis, and left parenthesis 4 comma 2 right parenthesis.
Image with alt text: A coordinate plane shows the x-axis ranging from negative 9 to 9 in increments of 1 and the y-axis ranging from 11 to negative 11 in increments of 1. A line with arrows at both ends joins two plotted points. The coordinates of the plotted points are as follows: left parenthesis 0 comma 3 right parenthesis, and left parenthesis 4 comma 2 right parenthesis.

A coordinate plane shows the x-axis ranging from negative 9 to 9 in increments of 1 and the y-axis ranging from 11 to negative 11 in increments of 1. A line with arrows at both ends joins two plotted points. The coordinates of the plotted points are as follows: left parenthesis 0 comma negative 3 right parenthesis, and left parenthesis 7 comma negative 1 right parenthesis.
Image with alt text: A coordinate plane shows the x-axis ranging from negative 9 to 9 in increments of 1 and the y-axis ranging from 11 to negative 11 in increments of 1. A line with arrows at both ends joins two plotted points. The coordinates of the plotted points are as follows: left parenthesis 0 comma negative 3 right parenthesis, and left parenthesis 7 comma negative 1 right parenthesis.

A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 2 comma 1 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 2 comma 1 right parenthesis.

Answer 1

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 \(y\):

1. Move \(9\) and the \(-x\) to the right side:
\[
\frac{3}{2}y = 9 - x
\]

2. 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:

1. Start at the y-intercept \( (0, 6) \) on the y-axis.
2. 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 \( (3, 4) \).
3. 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.