Sure, let's work through this problem step-by-step.
1. Change the equation into slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
\begin{array}{l}
3y = 1470 - 2x
\end{array}
\][/tex]
First, solve for [tex]\(y\)[/tex] by isolating [tex]\(y\)[/tex] on one side of the equation:
[tex]\[
\begin{array}{l}
3y = 1470 - 2x \\
y = \frac{1470}{3} - \frac{2x}{3} \\
y = 490 - \frac{2}{3}x
\end{array}
\][/tex]
Rewriting it in the standard form [tex]\(y = mx + b\)[/tex], we get:
[tex]\[
y = -\frac{2}{3}x + 490
\][/tex]
From the final form of the equation [tex]\(y = mx + b\)[/tex], we identify:
- The slope ([tex]\(m\)[/tex]) is [tex]\(-\frac{2}{3}\)[/tex].
- The y-intercept ([tex]\(b\)[/tex]) is [tex]\(490\)[/tex].
2. Describe how to graph the line using the slope-intercept method:
To graph the line using the slope-intercept method, follow these steps:
- Step 1: Start by plotting the y-intercept. The y-intercept is the point where the line crosses the y-axis. In this case, the y-intercept [tex]\(b\)[/tex] is [tex]\(490\)[/tex]. So, plot the point [tex]\((0, 490)\)[/tex] on the graph.
- Step 2: Use the slope to determine the next point. The slope [tex]\(m\)[/tex] of [tex]\(-\frac{2}{3}\)[/tex] means that for every 3 units you move horizontally to the right (positive direction of [tex]\(x\)[/tex]), you move 2 units down (negative direction of [tex]\(y\)[/tex]). Conversely, for every 3 units you move to the left, you move 2 units up.
- From the point [tex]\((0, 490)\)[/tex], move 3 units to the right along the x-axis to [tex]\(x = 3\)[/tex], then move 2 units down along the y-axis to get to the point [tex]\((3, 488)\)[/tex].
- Alternatively, you can move 3 units to the left along the x-axis to [tex]\(x = -3\)[/tex], then move 2 units up along the y-axis to get to the point [tex]\((-3, 492)\)[/tex].
- Step 3: Plot these points on the graph and draw a straight line through them. Make sure the line extends across the entire graph.
By following these steps, you have correctly graphed the line using the slope-intercept method. The line will have a negative slope, gently decreasing from left to right, reflecting the slope of [tex]\(-\frac{2}{3}\)[/tex].
