Question
Ms. Mac asked her students to determine a solution for 2x - 5y = 9. Bella believes (-2, 1) is a solution to the equation. Marco believes (11, 2) is a solution to the equation. Liam believes (2, -1) is a solution to the equation.
Answers
GPT-4o mini
To determine if each of the provided points is a solution to the equation \(2x - 5y = 9\), we will substitute the \(x\) and \(y\) values from each point into the equation and see if the equation holds true.
1. **For Bella's point (-2, 1):**
\[
x = -2, \quad y = 1
\]
Substitute into the equation:
\[
2(-2) - 5(1) = -4 - 5 = -9
\]
Since \(-9 \neq 9\), Bella's point is **not a solution**.
2. **For Marco's point (11, 2):**
\[
x = 11, \quad y = 2
\]
Substitute into the equation:
\[
2(11) - 5(2) = 22 - 10 = 12
\]
Since \(12 \neq 9\), Marco's point is **not a solution**.
3. **For Liam's point (2, -1):**
\[
x = 2, \quad y = -1
\]
Substitute into the equation:
\[
2(2) - 5(-1) = 4 + 5 = 9
\]
Since \(9 = 9\), Liam's point is **a solution**.
In summary:
- Bella's point (-2, 1) is not a solution.
- Marco's point (11, 2) is not a solution.
- Liam's point (2, -1) is a solution.
1. **For Bella's point (-2, 1):**
\[
x = -2, \quad y = 1
\]
Substitute into the equation:
\[
2(-2) - 5(1) = -4 - 5 = -9
\]
Since \(-9 \neq 9\), Bella's point is **not a solution**.
2. **For Marco's point (11, 2):**
\[
x = 11, \quad y = 2
\]
Substitute into the equation:
\[
2(11) - 5(2) = 22 - 10 = 12
\]
Since \(12 \neq 9\), Marco's point is **not a solution**.
3. **For Liam's point (2, -1):**
\[
x = 2, \quad y = -1
\]
Substitute into the equation:
\[
2(2) - 5(-1) = 4 + 5 = 9
\]
Since \(9 = 9\), Liam's point is **a solution**.
In summary:
- Bella's point (-2, 1) is not a solution.
- Marco's point (11, 2) is not a solution.
- Liam's point (2, -1) is a solution.
T-Swizzle
is Bella and Marcos solution below or above the graph
GPT-4o mini
To determine whether Bella's and Marco's points lie above or below the graph of the equation \(2x - 5y = 9\), we can evaluate the equation at their \(x\) values and compare the corresponding \(y\) values.
1. **For Bella's point (-2, 1):**
- Calculate the \(y\) value on the line when \(x = -2\):
\[
2(-2) - 5y = 9 \implies -4 - 5y = 9 \implies -5y = 13 \implies y = -\frac{13}{5} \approx -2.6
\]
- Bella's \(y\) value is 1, while the calculated \(y\) value on the line is approximately -2.6.
- Since \(1 > -2.6\), Bella's point (-2, 1) lies **above** the graph of the equation.
2. **For Marco's point (11, 2):**
- Calculate the \(y\) value on the line when \(x = 11\):
\[
2(11) - 5y = 9 \implies 22 - 5y = 9 \implies -5y = -13 \implies y = \frac{13}{5} \approx 2.6
\]
- Marco's \(y\) value is 2, while the calculated \(y\) value on the line is approximately 2.6.
- Since \(2 < 2.6\), Marco's point (11, 2) lies **below** the graph of the equation.
In summary:
- Bella's point (-2, 1) is **above** the graph.
- Marco's point (11, 2) is **below** the graph.
1. **For Bella's point (-2, 1):**
- Calculate the \(y\) value on the line when \(x = -2\):
\[
2(-2) - 5y = 9 \implies -4 - 5y = 9 \implies -5y = 13 \implies y = -\frac{13}{5} \approx -2.6
\]
- Bella's \(y\) value is 1, while the calculated \(y\) value on the line is approximately -2.6.
- Since \(1 > -2.6\), Bella's point (-2, 1) lies **above** the graph of the equation.
2. **For Marco's point (11, 2):**
- Calculate the \(y\) value on the line when \(x = 11\):
\[
2(11) - 5y = 9 \implies 22 - 5y = 9 \implies -5y = -13 \implies y = \frac{13}{5} \approx 2.6
\]
- Marco's \(y\) value is 2, while the calculated \(y\) value on the line is approximately 2.6.
- Since \(2 < 2.6\), Marco's point (11, 2) lies **below** the graph of the equation.
In summary:
- Bella's point (-2, 1) is **above** the graph.
- Marco's point (11, 2) is **below** the graph.