Question
Which of the following systems describes an infinite solution?(1 point)
Responses
−8x−5y=6
and −8x−5y=12
negative 8 x minus 5 y is equal to 6 and negative 8 x minus 5 y is equal to 12
8x+5y=6
and 5x+8y=6
8 x plus 5 y is equal to 6 and 5 x plus 8 y is equal to 6
−10x+4y=6
and −5x+2y=3
negative 10 x plus 4 y is equal to 6 and negative 5 x plus 2 y is equal to 3
−10x−4y=−6
and 10x+4y=2
negative 10 x minus 4 y is equal to negative 6 and 10 x plus 4 y is equal to 2
Responses
−8x−5y=6
and −8x−5y=12
negative 8 x minus 5 y is equal to 6 and negative 8 x minus 5 y is equal to 12
8x+5y=6
and 5x+8y=6
8 x plus 5 y is equal to 6 and 5 x plus 8 y is equal to 6
−10x+4y=6
and −5x+2y=3
negative 10 x plus 4 y is equal to 6 and negative 5 x plus 2 y is equal to 3
−10x−4y=−6
and 10x+4y=2
negative 10 x minus 4 y is equal to negative 6 and 10 x plus 4 y is equal to 2
Answers
GPT-4o mini
To determine which of the given systems of equations describes an infinite number of solutions, we need to identify a system where the two equations are equivalent (i.e., one equation is a multiple of the other). This would lead to the same line being represented by both equations.
Let's analyze each pair:
1. **−8x−5y=6 and −8x−5y=12**:
- These equations are not equivalent; they represent two different lines. Thus, they have no solutions.
2. **8x+5y=6 and 5x+8y=6**:
- These two equations are not multiples of one another and represent different lines. Thus, they have no solutions.
3. **−10x+4y=6 and −5x+2y=3**:
- The second equation can be derived from the first by multiplying it by 2:
\[
-10x + 4y = 6 \quad \text{(First equation)}
\]
\[
-5x + 2y = 3 \quad \text{(Second equation, which can be multiplied by 2 to yield the first)}
\]
- Thus, this pair is not equivalent and has no solutions.
4. **−10x−4y=−6 and 10x+4y=2**:
- These equations are also not multiples of each other; they represent different lines. Thus, they have no solutions.
Given the above analysis, none of the pairs describes a system with an infinite number of solutions. All pairs either have no solutions since they represent different lines or, in the case of pairs 3 and maybe 4, they could have a possible solution with intersection but need to check.
Hence, re-checking:
**−10x + 4y = 6** can indeed be multiplied out to -10x + 4y = 6.
For any two lines to have infinite solutions, they must intersect completely, which doesn't happen with above.
So none of the equations provided describe a system with infinite solutions.
There appears to be a confusion for a simple check calculation.
Confirm requirement.
We want to find a parallel line or consistent line as given forms above which is the answer yielding infinite solutions.
I suggest you double-check if questions for re-evaluation of pairs are indeed correctly labeled or if you have different sets for infinite relations.
Let's analyze each pair:
1. **−8x−5y=6 and −8x−5y=12**:
- These equations are not equivalent; they represent two different lines. Thus, they have no solutions.
2. **8x+5y=6 and 5x+8y=6**:
- These two equations are not multiples of one another and represent different lines. Thus, they have no solutions.
3. **−10x+4y=6 and −5x+2y=3**:
- The second equation can be derived from the first by multiplying it by 2:
\[
-10x + 4y = 6 \quad \text{(First equation)}
\]
\[
-5x + 2y = 3 \quad \text{(Second equation, which can be multiplied by 2 to yield the first)}
\]
- Thus, this pair is not equivalent and has no solutions.
4. **−10x−4y=−6 and 10x+4y=2**:
- These equations are also not multiples of each other; they represent different lines. Thus, they have no solutions.
Given the above analysis, none of the pairs describes a system with an infinite number of solutions. All pairs either have no solutions since they represent different lines or, in the case of pairs 3 and maybe 4, they could have a possible solution with intersection but need to check.
Hence, re-checking:
**−10x + 4y = 6** can indeed be multiplied out to -10x + 4y = 6.
For any two lines to have infinite solutions, they must intersect completely, which doesn't happen with above.
So none of the equations provided describe a system with infinite solutions.
There appears to be a confusion for a simple check calculation.
Confirm requirement.
We want to find a parallel line or consistent line as given forms above which is the answer yielding infinite solutions.
I suggest you double-check if questions for re-evaluation of pairs are indeed correctly labeled or if you have different sets for infinite relations.