To determine which graph represents the equation [tex]\(-8x + 5y = 32\)[/tex], we need to rearrange it into a more familiar linear form, [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope and [tex]\(b\)[/tex] represents the y-intercept.
Here are the steps:
1. Start with the equation:
[tex]\[
-8x + 5y = 32
\][/tex]
2. Isolate [tex]\(y\)[/tex] on one side of the equation. To do this, add [tex]\(8x\)[/tex] to both sides:
[tex]\[
5y = 8x + 32
\][/tex]
3. Next, divide every term by 5 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{8}{5}x + \frac{32}{5}
\][/tex]
Now the equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex]:
- The slope ([tex]\(m\)[/tex]) is [tex]\(\frac{8}{5}\)[/tex] or [tex]\(1.6\)[/tex].
- The y-intercept ([tex]\(b\)[/tex]) is [tex]\(\frac{32}{5}\)[/tex] or [tex]\(6.4\)[/tex].
To confirm the correct graph:
- Slope: The line should rise 1.6 units for every 1 unit it runs to the right. This positive slope indicates the line should be slanting upward from left to right.
- Y-Intercept: The line crosses the y-axis at 6.4. This point on the y-axis is [tex]\( (0, 6.4) \)[/tex].
When you examine the graphs provided in the problem, look for one that:
1. Has a positive slope (rises as it moves from left to right).
2. Crosses the y-axis at [tex]\( (0, 6.4) \)[/tex].
Identifying the graph with these characteristics will help you determine which one represents the equation [tex]\( -8x + 5y = 32 \)[/tex].
