Question

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

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.

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.

Answer 1

To graph the equation \(9 - x = \frac{3}{2}y\) in slope-intercept form, we first need to rearrange the equation into the format \(y = mx + b\).

Starting from the original 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 \(\frac{2}{3}\):
\[
y = \frac{2}{3}(9) - \frac{2}{3}x
\]
\[
y = 6 - \frac{2}{3}x
\]

Now, we can rewrite this in slope-intercept form as:
\[
y = -\frac{2}{3}x + 6
\]

Here, the slope \(m = -\frac{2}{3}\) and the y-intercept \(b = 6\). This means the line crosses the y-axis at 6 and has a slope indicating it decreases by \(\frac{2}{3}\) for every unit increase in \(x\).

Let's find two points to plot:
1. The y-intercept: when \(x = 0\), \(y = 6\) → point \((0, 6)\)
2. Another point: Let's choose \(x = 3\):
\[
y = -\frac{2}{3}(3) + 6 = -2 + 6 = 4
\]
→ point \((3, 4)\)

Using these points \((0, 6)\) and \((3, 4)\), we can graph the line.

Based on the options presented, the second illustration, which shows a downward slanting line that passes through the plotted points \((0, 6)\) and \((3, 4)\), is the correct graph of the equation \(9 - x = \frac{3}{2}y\).