Sure! We are given the equation:
[tex]\[ 6x + 9y = 1500 \][/tex]
To find the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we can rearrange this equation to express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
First, we isolate [tex]\( y \)[/tex]:
1. Subtract [tex]\( 6x \)[/tex] from both sides of the equation:
[tex]\[ 9y = 1500 - 6x \][/tex]
2. Divide both sides by 9 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1500 - 6x}{9} \][/tex]
[tex]\[ y = \frac{1500}{9} - \frac{6x}{9} \][/tex]
[tex]\[ y = \frac{1500}{9} - \frac{2x}{3} \][/tex]
Now, simplifying [tex]\( \frac{1500}{9} \)[/tex]:
[tex]\[ y = 166.67 - \frac{2x}{3} \][/tex]
This is the equation of a straight line in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Here, the slope [tex]\( m \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex] and the y-intercept [tex]\( b \)[/tex] is 166.67.
Next, we can plot this relationship on graph paper.
1. Begin by marking the y-intercept (which is 166.67) on the y-axis.
2. The slope [tex]\( -\frac{2}{3} \)[/tex] tells us that for every 3 units we move to the right on the x-axis, we move down 2 units on the y-axis.
To give more precision to our plot, let's find the x-intercept (where [tex]\( y = 0 \)[/tex]):
From [tex]\( y = 166.67 - \frac{2x}{3} \)[/tex],
set [tex]\( y = 0 \)[/tex]:
[tex]\[ 0 = 166.67 - \frac{2x}{3} \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ \frac{2x}{3} = 166.67 \][/tex]
[tex]\[ 2x = 166.67 \times 3 \][/tex]
[tex]\[ 2x = 500 \][/tex]
[tex]\[ x = 250 \][/tex]
The x-intercept is 250.
We now have two key points to plot: [tex]\( (0, 166.67) \)[/tex] and [tex]\( (250, 0) \)[/tex].
Let's plot these points and draw the straight line passing through them:
1. Point (0, 166.67): plot on the y-axis.
2. Point (250, 0): plot on the x-axis.
Draw a line connecting these two points, extending the line across the axes. The graph of the equation [tex]\( 6x + 9y = 1500 \)[/tex] is a straight line that will cross the y-axis at 166.67 and the x-axis at 250.
The graph should look like this:
[tex]\[
\begin{array}{c}
y \\
| \\
| \\
| \\
| \;\;(0, 166.67) \\
| \\
| \\
| \\
| \\
| \\
| ------------------------------- x \\
(250, 0)
\end{array}
\][/tex]
This straight line represents the relationship [tex]\( 6x + 9y = 1500 \)[/tex] between the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
