Question
Solve each system of equations on the left. Match the system to the correct number of solutions.(3 points)
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2x-2y=-4
y=x+2
y=3x-2
2x+3y=16
y=-2x-7
y=-2x+3
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
2x-2y=-4
y=x+2
y=3x-2
2x+3y=16
y=-2x-7
y=-2x+3
Answers
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Answered by
GPT-4o mini
To solve the given systems of equations, we will analyze each system and determine the number of solutions for each.
### System 1:
1. \( 2x - 2y = -4 \)
2. \( y = x + 2 \)
First, let's rearrange the first equation to solve for \(y\):
\[ 2x - 2y = -4 \]
\[ -2y = -4 - 2x \]
\[ 2y = 4 + 2x \]
\[ y = 2 + x \]
Now, comparing the two equations:
- From the first equation rearranged: \( y = x + 2 \)
- From the second equation: \( y = x + 2 \)
Since both lines are the same, there are infinitely many solutions.
### System 2:
1. \( y = 3x - 2 \)
2. \( 2x + 3y = 16 \)
Now we can substitute the first equation into the second:
- Substitute \( y \) in the second equation:
\[ 2x + 3(3x - 2) = 16 \]
\[ 2x + 9x - 6 = 16 \]
\[ 11x - 6 = 16 \]
\[ 11x = 22 \]
\[ x = 2 \]
Now substitute back to find \(y\):
\[ y = 3(2) - 2 = 6 - 2 = 4 \]
Thus, there is **one solution** at the point \( (2, 4) \).
### System 3:
1. \( y = -2x - 7 \)
2. \( y = -2x + 3 \)
Comparing the two equations:
- From the first equation: \( y = -2x - 7 \)
- From the second equation: \( y = -2x + 3 \)
Since the slopes (-2) are equal but the y-intercepts are different (-7 and +3), these lines are parallel. Thus, there is **no solution**.
### Summary:
- System 1: **Infinitely many solutions**
- System 2: **One solution**
- System 3: **No solution**
### Matching Systems to Number of Solutions:
- Infinitely many solutions: for \(2x - 2y = -4\) and \(y = x + 2\)
- One solution: for \(y = 3x - 2\) and \(2x + 3y = 16\)
- No solution: for \(y = -2x - 7\) and \(y = -2x + 3\)
Please match the systems accordingly in your input.
### System 1:
1. \( 2x - 2y = -4 \)
2. \( y = x + 2 \)
First, let's rearrange the first equation to solve for \(y\):
\[ 2x - 2y = -4 \]
\[ -2y = -4 - 2x \]
\[ 2y = 4 + 2x \]
\[ y = 2 + x \]
Now, comparing the two equations:
- From the first equation rearranged: \( y = x + 2 \)
- From the second equation: \( y = x + 2 \)
Since both lines are the same, there are infinitely many solutions.
### System 2:
1. \( y = 3x - 2 \)
2. \( 2x + 3y = 16 \)
Now we can substitute the first equation into the second:
- Substitute \( y \) in the second equation:
\[ 2x + 3(3x - 2) = 16 \]
\[ 2x + 9x - 6 = 16 \]
\[ 11x - 6 = 16 \]
\[ 11x = 22 \]
\[ x = 2 \]
Now substitute back to find \(y\):
\[ y = 3(2) - 2 = 6 - 2 = 4 \]
Thus, there is **one solution** at the point \( (2, 4) \).
### System 3:
1. \( y = -2x - 7 \)
2. \( y = -2x + 3 \)
Comparing the two equations:
- From the first equation: \( y = -2x - 7 \)
- From the second equation: \( y = -2x + 3 \)
Since the slopes (-2) are equal but the y-intercepts are different (-7 and +3), these lines are parallel. Thus, there is **no solution**.
### Summary:
- System 1: **Infinitely many solutions**
- System 2: **One solution**
- System 3: **No solution**
### Matching Systems to Number of Solutions:
- Infinitely many solutions: for \(2x - 2y = -4\) and \(y = x + 2\)
- One solution: for \(y = 3x - 2\) and \(2x + 3y = 16\)
- No solution: for \(y = -2x - 7\) and \(y = -2x + 3\)
Please match the systems accordingly in your input.
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