Question
Steve wants to create a system of equations so that the system has no solution. Which of these can Steve do? Check all that apply.
Create a pair of equations with the same slope and different y-intercepts.
Create a pair of lines in which one line lies directly on top of the other.
Create a pair of equivalent equations.
Create a pair of lines that will always stay the same distance apart.
Create a pair of lines that intersect at only one point.
Create a pair of equations with the same slope and different y-intercepts.
Create a pair of lines in which one line lies directly on top of the other.
Create a pair of equivalent equations.
Create a pair of lines that will always stay the same distance apart.
Create a pair of lines that intersect at only one point.
Answers
GPT-4o mini
To create a system of equations that has no solution, the key is to have two lines that are parallel, meaning they have the same slope but different y-intercepts. Let's evaluate each option:
1. **Create a pair of equations with the same slope and different y-intercepts.**
- This will create parallel lines, which means there will be no solution. **This option is valid.**
2. **Create a pair of lines in which one line lies directly on top of the other.**
- This situation describes identical lines, which means that there are infinitely many solutions. **This option is not valid.**
3. **Create a pair of equivalent equations.**
- Equivalent equations represent the same line, thus providing infinitely many solutions. **This option is not valid.**
4. **Create a pair of lines that will always stay the same distance apart.**
- This describes parallel lines, which have no solutions. **This option is valid.**
5. **Create a pair of lines that intersect at only one point.**
- Lines that intersect at one point have exactly one solution. **This option is not valid.**
Based on this evaluation, the valid options for Steve are:
- Create a pair of equations with the same slope and different y-intercepts.
- Create a pair of lines that will always stay the same distance apart.
1. **Create a pair of equations with the same slope and different y-intercepts.**
- This will create parallel lines, which means there will be no solution. **This option is valid.**
2. **Create a pair of lines in which one line lies directly on top of the other.**
- This situation describes identical lines, which means that there are infinitely many solutions. **This option is not valid.**
3. **Create a pair of equivalent equations.**
- Equivalent equations represent the same line, thus providing infinitely many solutions. **This option is not valid.**
4. **Create a pair of lines that will always stay the same distance apart.**
- This describes parallel lines, which have no solutions. **This option is valid.**
5. **Create a pair of lines that intersect at only one point.**
- Lines that intersect at one point have exactly one solution. **This option is not valid.**
Based on this evaluation, the valid options for Steve are:
- Create a pair of equations with the same slope and different y-intercepts.
- Create a pair of lines that will always stay the same distance apart.