Question
Match the system of equations to the number of solutions it has.
Put responses in the correct input to answer the question.
1. y = 5x + 7
3y - 15x = 21
2. y = -13x - 4
y = 3x + 6
3. y = 5x - 1
y = 5x + 2
Put responses in the correct input to answer the question.
1. y = 5x + 7
3y - 15x = 21
2. y = -13x - 4
y = 3x + 6
3. y = 5x - 1
y = 5x + 2
Answers
GPT-4o mini
Let's analyze each system of equations to determine the number of solutions:
1. **First system:**
- Equation 1: \( y = 5x + 7 \)
- Equation 2: \( 3y - 15x = 21 \)
- Simplifying Equation 2:
\[ 3y = 15x + 21 \]
\[ y = 5x + 7 \]
- This means both equations represent the same line, so there are infinitely many solutions.
2. **Second system:**
- Equation 1: \( y = -13x - 4 \)
- Equation 2: \( y = 3x + 6 \)
- These are two lines with different slopes (the first has a slope of -13 and the second has a slope of 3), so they will intersect at exactly one point, resulting in one solution.
3. **Third system:**
- Equation 1: \( y = 5x - 1 \)
- Equation 2: \( y = 5x + 2 \)
- Both lines have the same slope (5), but different y-intercepts (-1 and +2), indicating they are parallel and will never intersect. Thus, there are no solutions.
Now, summarizing the findings:
- The first system has **infinitely many solutions.**
- The second system has **one solution.**
- The third system has **no solutions.**
Here’s the final arrangement of answers based on the systems:
1. Infinitely many solutions
2. One solution
3. No solutions
1. **First system:**
- Equation 1: \( y = 5x + 7 \)
- Equation 2: \( 3y - 15x = 21 \)
- Simplifying Equation 2:
\[ 3y = 15x + 21 \]
\[ y = 5x + 7 \]
- This means both equations represent the same line, so there are infinitely many solutions.
2. **Second system:**
- Equation 1: \( y = -13x - 4 \)
- Equation 2: \( y = 3x + 6 \)
- These are two lines with different slopes (the first has a slope of -13 and the second has a slope of 3), so they will intersect at exactly one point, resulting in one solution.
3. **Third system:**
- Equation 1: \( y = 5x - 1 \)
- Equation 2: \( y = 5x + 2 \)
- Both lines have the same slope (5), but different y-intercepts (-1 and +2), indicating they are parallel and will never intersect. Thus, there are no solutions.
Now, summarizing the findings:
- The first system has **infinitely many solutions.**
- The second system has **one solution.**
- The third system has **no solutions.**
Here’s the final arrangement of answers based on the systems:
1. Infinitely many solutions
2. One solution
3. No solutions