Question
what are the graph points of this equation 10.5x + 5.5y + 150 is less than or equal to 300
no decimals or negative numbers allowed
no decimals or negative numbers allowed
Answers
GPT-4o mini
To analyze the inequality \( 10.5x + 5.5y + 150 \leq 300 \) without using decimals or negative numbers, we can first simplify the inequality.
1. **Convert the Inequality**:
\[
10.5x + 5.5y + 150 \leq 300
\]
Subtract 150 from both sides:
\[
10.5x + 5.5y \leq 150
\]
2. **Eliminate Decimals**:
To eliminate decimals, multiply the entire inequality by 2 (which is the least number to make coefficients whole numbers):
\[
2(10.5x) + 2(5.5y) \leq 2(150)
\]
Simplifying gives us:
\[
21x + 11y \leq 300
\]
3. **Find the Boundary Points**:
To find the points that satisfy the equation \( 21x + 11y = 300 \), we can find the intercepts where \( x = 0 \) and \( y = 0 \).
- **When \( x = 0 \)**:
\[
11y = 300 \quad \Rightarrow \quad y = \frac{300}{11} \approx 27.27
\]
Taking only whole numbers gives \( y = 27 \) when rounded down, so the point (0, 27).
- **When \( y = 0 \)**:
\[
21x = 300 \quad \Rightarrow \quad x = \frac{300}{21} \approx 14.29
\]
Taking only whole numbers gives \( x = 14 \) when rounded down, so the point (14, 0).
4. **Check for Other Integer Points**:
We can find integer solutions (x, y) to the equation for values ranging from \( x = 0 \) to \( x = 14 \):
- For \( x = 0 \): \( 21(0) + 11y \leq 300 \) → \( y \) can be from 0 to 27.
- For \( x = 1 \): \( 21(1) + 11y \leq 300 \) → \( 11y \leq 279 \) → \( y \) can be from 0 to 25.
- For \( x = 2 \): \( 21(2) + 11y \leq 300 \) → \( 11y \leq 258 \) → \( y \) can be from 0 to 23.
- Continue this process until \( x = 14 \):
- \( x = 14 \): \( 21(14) + 11y \leq 300 \) → \( 11y \leq 0 \) → \( y = 0 \).
5. **Summary of Integer Points**:
From these calculations, valid integer points (x,y) where both \( x \) and \( y \) are non-negative integers include:
- For \( x = 0 \): Points (0, 0) to (0, 27)
- For \( x = 1 \): Points (1, 0) to (1, 25)
- For \( x = 2 \): Points (2, 0) to (2, 23)
- For \( x = 3 \): Points (3, 0) to (3, 22)
- For \( x = 4 \): Points (4, 0) to (4, 20)
- For \( x = 5 \): Points (5, 0) to (5, 19)
- For \( x = 6 \): Points (6, 0) to (6, 17)
- For \( x = 7 \): Points (7, 0) to (7, 16)
- For \( x = 8 \): Points (8, 0) to (8, 14)
- For \( x = 9 \): Points (9, 0) to (9, 13)
- For \( x = 10 \): Points (10, 0) to (10, 11)
- For \( x = 11 \): Points (11, 0) to (11, 10)
- For \( x = 12 \): Points (12, 0) to (12, 8)
- For \( x = 13 \): Points (13, 0) to (13, 7)
- For \( x = 14 \): Point (14, 0)
These points represent all integer solutions for the inequality \( 21x + 11y \leq 300 \) without decimals or negative numbers.
1. **Convert the Inequality**:
\[
10.5x + 5.5y + 150 \leq 300
\]
Subtract 150 from both sides:
\[
10.5x + 5.5y \leq 150
\]
2. **Eliminate Decimals**:
To eliminate decimals, multiply the entire inequality by 2 (which is the least number to make coefficients whole numbers):
\[
2(10.5x) + 2(5.5y) \leq 2(150)
\]
Simplifying gives us:
\[
21x + 11y \leq 300
\]
3. **Find the Boundary Points**:
To find the points that satisfy the equation \( 21x + 11y = 300 \), we can find the intercepts where \( x = 0 \) and \( y = 0 \).
- **When \( x = 0 \)**:
\[
11y = 300 \quad \Rightarrow \quad y = \frac{300}{11} \approx 27.27
\]
Taking only whole numbers gives \( y = 27 \) when rounded down, so the point (0, 27).
- **When \( y = 0 \)**:
\[
21x = 300 \quad \Rightarrow \quad x = \frac{300}{21} \approx 14.29
\]
Taking only whole numbers gives \( x = 14 \) when rounded down, so the point (14, 0).
4. **Check for Other Integer Points**:
We can find integer solutions (x, y) to the equation for values ranging from \( x = 0 \) to \( x = 14 \):
- For \( x = 0 \): \( 21(0) + 11y \leq 300 \) → \( y \) can be from 0 to 27.
- For \( x = 1 \): \( 21(1) + 11y \leq 300 \) → \( 11y \leq 279 \) → \( y \) can be from 0 to 25.
- For \( x = 2 \): \( 21(2) + 11y \leq 300 \) → \( 11y \leq 258 \) → \( y \) can be from 0 to 23.
- Continue this process until \( x = 14 \):
- \( x = 14 \): \( 21(14) + 11y \leq 300 \) → \( 11y \leq 0 \) → \( y = 0 \).
5. **Summary of Integer Points**:
From these calculations, valid integer points (x,y) where both \( x \) and \( y \) are non-negative integers include:
- For \( x = 0 \): Points (0, 0) to (0, 27)
- For \( x = 1 \): Points (1, 0) to (1, 25)
- For \( x = 2 \): Points (2, 0) to (2, 23)
- For \( x = 3 \): Points (3, 0) to (3, 22)
- For \( x = 4 \): Points (4, 0) to (4, 20)
- For \( x = 5 \): Points (5, 0) to (5, 19)
- For \( x = 6 \): Points (6, 0) to (6, 17)
- For \( x = 7 \): Points (7, 0) to (7, 16)
- For \( x = 8 \): Points (8, 0) to (8, 14)
- For \( x = 9 \): Points (9, 0) to (9, 13)
- For \( x = 10 \): Points (10, 0) to (10, 11)
- For \( x = 11 \): Points (11, 0) to (11, 10)
- For \( x = 12 \): Points (12, 0) to (12, 8)
- For \( x = 13 \): Points (13, 0) to (13, 7)
- For \( x = 14 \): Point (14, 0)
These points represent all integer solutions for the inequality \( 21x + 11y \leq 300 \) without decimals or negative numbers.