Question

what are the plot points of this equation 10.5x + 5.5y + 150 is less than or equal to 300

Answer 1

To analyze the inequality \(10.5x + 5.5y + 150 \leq 300\), first, we can simplify it by isolating the terms involving \(x\) and \(y\).

1. Subtract 150 from both sides:
\[
10.5x + 5.5y \leq 150
\]

2. It can be helpful to rewrite this in the standard form of a linear equation, which helps in understanding the related boundary line. We can rearrange it:
\[
5.5y \leq -10.5x + 150
\]
Or:
\[
y \leq -\frac{10.5}{5.5}x + \frac{150}{5.5}
\]
Simplifying further, we calculate:
\[
-\frac{10.5}{5.5} \approx -1.9091 \quad \text{and} \quad \frac{150}{5.5} \approx 27.2727
\]
So, the equation for the boundary line becomes:
\[
y = -1.9091x + 27.2727
\]

3. To find specific plot points that satisfy the inequality, we can calculate the corresponding \(y\) values for different \(x\) values along the boundary line.

- When \(x = 0\):
\[
y = 27.2727
\]
(Point: \( (0, 27.2727) \))

- When \(x = 10\):
\[
y = -1.9091(10) + 27.2727 = -19.091 + 27.2727 \approx 8.1817
\]
(Point: \( (10, 8.1817) \))

- When \(x = 20\):
\[
y = -1.9091(20) + 27.2727 = -38.182 + 27.2727 \approx -10.9093
\]
(Point: \( (20, -10.9093) \))

4. The inequality signifies that we are interested in the region below (and including) the boundary line.

### Summary of Plot Points
1. \( (0, 27.2727) \)
2. \( (10, 8.1817) \)
3. \( (20, -10.9093) \)

### Graphing
- On a graph, plot the line \(y = -1.9091x + 27.2727\).
- Shade the area below this line where the inequality \(y \leq -1.9091x + 27.2727\) holds true. The boundary line itself is included in the solution (solid line) because of the "less than or equal to" part of the inequality.

You can choose any \(x\) values within the constraints of the context to find additional \(y\) values, as long as you satisfy the inequality!