Question
The point (2,5) is a solution to which system of equations?(1 point)
Responses
y = x -8
2x+y=7
y=-1/2x+6
y=3x-1
y=x+2
y=x+5
y=2/3x + 6
3y+6x-18=0
Responses
y = x -8
2x+y=7
y=-1/2x+6
y=3x-1
y=x+2
y=x+5
y=2/3x + 6
3y+6x-18=0
Answers
Answer
Which one
Answered by
GPT-4o mini
To determine which system of equations the point (2, 5) is a solution for, we need to substitute \(x = 2\) and \(y = 5\) into each system of equations and check if they hold true.
1. **First system:**
- \(y = x - 8\)
- \(5 = 2 - 8\) ⇒ \(5 = -6\) (False)
- \(2x + y = 7\)
- \(2(2) + 5 = 7\) ⇒ \(4 + 5 = 7\) ⇒ \(9 = 7\) (False)
2. **Second system:**
- \(y = -\frac{1}{2}x + 6\)
- \(5 = -\frac{1}{2}(2) + 6\) ⇒ \(5 = -1 + 6\) ⇒ \(5 = 5\) (True)
- \(y = 3x - 1\)
- \(5 = 3(2) - 1\) ⇒ \(5 = 6 - 1\) ⇒ \(5 = 5\) (True)
Since both equations are satisfied, (2, 5) is a solution to this system.
3. **Third system:**
- \(y = x + 2\)
- \(5 = 2 + 2\) ⇒ \(5 = 4\) (False)
- \(y = x + 5\)
- \(5 = 2 + 5\) ⇒ \(5 = 7\) (False)
4. **Fourth system:**
- \(y = \frac{2}{3}x + 6\)
- \(5 = \frac{2}{3}(2) + 6\) ⇒ \(5 = \frac{4}{3} + 6\) ⇒ \(5 \neq \frac{22}{3}\) (False)
- \(3y + 6x - 18 = 0\)
- \(3(5) + 6(2) - 18 = 0\) ⇒ \(15 + 12 - 18 = 0\) ⇒ \(9 \neq 0\) (False)
The only system where both equations are satisfied by the point (2, 5) is the **second system**:
\[
y = -\frac{1}{2}x + 6
\]
\[
y = 3x - 1
\]
So, (2, 5) is a solution to the second system of equations.
1. **First system:**
- \(y = x - 8\)
- \(5 = 2 - 8\) ⇒ \(5 = -6\) (False)
- \(2x + y = 7\)
- \(2(2) + 5 = 7\) ⇒ \(4 + 5 = 7\) ⇒ \(9 = 7\) (False)
2. **Second system:**
- \(y = -\frac{1}{2}x + 6\)
- \(5 = -\frac{1}{2}(2) + 6\) ⇒ \(5 = -1 + 6\) ⇒ \(5 = 5\) (True)
- \(y = 3x - 1\)
- \(5 = 3(2) - 1\) ⇒ \(5 = 6 - 1\) ⇒ \(5 = 5\) (True)
Since both equations are satisfied, (2, 5) is a solution to this system.
3. **Third system:**
- \(y = x + 2\)
- \(5 = 2 + 2\) ⇒ \(5 = 4\) (False)
- \(y = x + 5\)
- \(5 = 2 + 5\) ⇒ \(5 = 7\) (False)
4. **Fourth system:**
- \(y = \frac{2}{3}x + 6\)
- \(5 = \frac{2}{3}(2) + 6\) ⇒ \(5 = \frac{4}{3} + 6\) ⇒ \(5 \neq \frac{22}{3}\) (False)
- \(3y + 6x - 18 = 0\)
- \(3(5) + 6(2) - 18 = 0\) ⇒ \(15 + 12 - 18 = 0\) ⇒ \(9 \neq 0\) (False)
The only system where both equations are satisfied by the point (2, 5) is the **second system**:
\[
y = -\frac{1}{2}x + 6
\]
\[
y = 3x - 1
\]
So, (2, 5) is a solution to the second system of equations.
Answered by
GPT-4o mini
The point \((2, 5)\) is a solution to the following system of equations:
1. \(y = -\frac{1}{2}x + 6\)
2. \(y = 3x - 1\)
This is the second option from your list.
1. \(y = -\frac{1}{2}x + 6\)
2. \(y = 3x - 1\)
This is the second option from your list.
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