Question
Ms.Mac asked her students to determine a solution for 2x -5 =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
Blank named a point That
Is a solution that lies on the graph of the equation. Blank named a point that is not a solution and lives above the graft of the equation. The point name a point that is not a solution and lasts below the graph of the equation.
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
Blank named a point That
Is a solution that lies on the graph of the equation. Blank named a point that is not a solution and lives above the graft of the equation. The point name a point that is not a solution and lasts below the graph of the equation.
Answers
GPT-4o mini
Let's first solve the equation \( 2x - 5 = 9 \) to find the actual solution.
1. Add 5 to both sides:
\[
2x = 14
\]
2. Divide by 2:
\[
x = 7
\]
The point that corresponds to this solution is \( (7, y) \) for any value of \( y \). Since the equation is in terms of \( x \) only, we can say that any point on the line defined by this equation will have the x-coordinate of 7.
Now, let's evaluate the points proposed by the students:
1. **Bella's point: (-2,1)**
\[
2(-2) - 5 = -4 - 5 = -9 \quad \text{(not a solution)}
\]
2. **Marco's point: (11,2)**
\[
2(11) - 5 = 22 - 5 = 17 \quad \text{(not a solution)}
\]
3. **Liam's point: (2,-1)**
\[
2(2) - 5 = 4 - 5 = -1 \quad \text{(not a solution)}
\]
That means none of the students have suggested valid solutions.
Now, let's find points as requested:
1. **A point that is a solution:**
- One solution is \( (7, 0) \). This means when \( x=7 \), we could choose \( y \) to be 0 (any \( y \) works).
2. **A point that is not a solution and lies above the graph:**
- A point above the graph would have a y-value that is greater than what the equation provides for that x-value. For instance, you can take \( (7, 5) \):
\[
2(7) - 5 = 14 - 5 = 9 \quad \text{(y-value at x=7)}
\]
So \( (7, 5) \) is above the graph since 5 is greater than 0.
3. **A point that is not a solution and lies below the graph:**
- A point below the graph could be \( (7, -1) \):
\[
2(7) - 5 = 14 - 5 = 9 \quad \text{(y-value at x=7)}
\]
Here, \( (7, -1) \) is below the graph since -1 is less than 0.
To summarize:
- A point that is a solution: **(7, 0)**
- A point that is not a solution and lies above the graph: **(7, 5)**
- A point that is not a solution and lies below the graph: **(7, -1)**
1. Add 5 to both sides:
\[
2x = 14
\]
2. Divide by 2:
\[
x = 7
\]
The point that corresponds to this solution is \( (7, y) \) for any value of \( y \). Since the equation is in terms of \( x \) only, we can say that any point on the line defined by this equation will have the x-coordinate of 7.
Now, let's evaluate the points proposed by the students:
1. **Bella's point: (-2,1)**
\[
2(-2) - 5 = -4 - 5 = -9 \quad \text{(not a solution)}
\]
2. **Marco's point: (11,2)**
\[
2(11) - 5 = 22 - 5 = 17 \quad \text{(not a solution)}
\]
3. **Liam's point: (2,-1)**
\[
2(2) - 5 = 4 - 5 = -1 \quad \text{(not a solution)}
\]
That means none of the students have suggested valid solutions.
Now, let's find points as requested:
1. **A point that is a solution:**
- One solution is \( (7, 0) \). This means when \( x=7 \), we could choose \( y \) to be 0 (any \( y \) works).
2. **A point that is not a solution and lies above the graph:**
- A point above the graph would have a y-value that is greater than what the equation provides for that x-value. For instance, you can take \( (7, 5) \):
\[
2(7) - 5 = 14 - 5 = 9 \quad \text{(y-value at x=7)}
\]
So \( (7, 5) \) is above the graph since 5 is greater than 0.
3. **A point that is not a solution and lies below the graph:**
- A point below the graph could be \( (7, -1) \):
\[
2(7) - 5 = 14 - 5 = 9 \quad \text{(y-value at x=7)}
\]
Here, \( (7, -1) \) is below the graph since -1 is less than 0.
To summarize:
- A point that is a solution: **(7, 0)**
- A point that is not a solution and lies above the graph: **(7, 5)**
- A point that is not a solution and lies below the graph: **(7, -1)**