Question
Fill in the blanks so that the following system of linear equations has infinite solutions.
Y=2x+7
Y=2x+7
Answers
GPT-4o mini
For a system of linear equations to have infinite solutions, both equations must represent the same line. This means that one equation can be derived from the other by scaling (multiplying or dividing by a nonzero constant).
Given the equation:
\[ y = 2x + 7 \]
If we let the second equation be:
\[ y = 2x + 7k \]
where \( k \) is a nonzero constant, the two equations will be equivalent for certain values of \( k \).
To illustrate, let's consider:
1. The first equation remains:
\[ y = 2x + 7 \]
2. For the second equation, let's pick \( k = 1 \) (which gives us the same equation), or generally, we can write it as:
\[ y = 2x + 7 \]
Thus, if we fill in the blanks of a second equation to ensure it's equivalent, we could have a second equation such as:
\[ 2y = 4x + 14 \]
This is simply a multiple of the first equation (in this case, multiplying the whole equation by 2). Both equations would then reflect the same line, giving us infinitely many solutions.
So, for infinite solutions, you can choose:
- Second equation: \( 2y = 4x + 14 \) or any other valid multiple that maintains the same relationship \( y = 2x + 7 \).
Given the equation:
\[ y = 2x + 7 \]
If we let the second equation be:
\[ y = 2x + 7k \]
where \( k \) is a nonzero constant, the two equations will be equivalent for certain values of \( k \).
To illustrate, let's consider:
1. The first equation remains:
\[ y = 2x + 7 \]
2. For the second equation, let's pick \( k = 1 \) (which gives us the same equation), or generally, we can write it as:
\[ y = 2x + 7 \]
Thus, if we fill in the blanks of a second equation to ensure it's equivalent, we could have a second equation such as:
\[ 2y = 4x + 14 \]
This is simply a multiple of the first equation (in this case, multiplying the whole equation by 2). Both equations would then reflect the same line, giving us infinitely many solutions.
So, for infinite solutions, you can choose:
- Second equation: \( 2y = 4x + 14 \) or any other valid multiple that maintains the same relationship \( y = 2x + 7 \).