Question
Fill in the blanks so that the following system of linear equations has infinite solutions.
y=-3x+4
y=-----x+-------
y=-3x+4
y=-----x+-------
Answers
GPT-4o mini
To ensure the system of linear equations has infinite solutions, the two equations must represent the same line. This means the second equation must be a multiple of the first equation.
The first equation is \( y = -3x + 4 \).
We can express the second equation in the form \( y = mx + b \), where \( m \) and \( b \) are to be determined.
If we multiply the entire first equation by a non-zero constant \( k \), we get:
\[
y = k(-3x + 4) = -3kx + 4k
\]
The second equation can then be written as:
\[
y = -3kx + 4k
\]
Comparing this with the given format \( y = \text{-----}x + \text{-------} \), we can assign:
- \( -3k \) as the coefficient of \( x \)
- \( 4k \) as the constant term
Thus, the filled-in version of the second equation can be:
\[
y = -3x + 4
\]
Here, if \( k = 1 \):
- The coefficient of \( x \) is \( -3 \) (leading to \( -3x \))
- The constant term is \( 4 \) (leading to \( +4\))
Therefore, one possible set of values can be:
\[
y = -3x + 4
\]
For example, if you let \( k = 2 \):
\[
y = -6x + 8
\]
Essentially, any \( k \neq 0 \) will work, provided you adjust both \( x \) and the constant term accordingly.
To summarize, the completed equation can be:
\[
y = -3x + 4 \quad \text{(for \( k = 1 \))}
\]
or, more generally:
\[
y = -3kx + 4k \quad \text{where \( k \) can be any non-zero constant.}
\]
The first equation is \( y = -3x + 4 \).
We can express the second equation in the form \( y = mx + b \), where \( m \) and \( b \) are to be determined.
If we multiply the entire first equation by a non-zero constant \( k \), we get:
\[
y = k(-3x + 4) = -3kx + 4k
\]
The second equation can then be written as:
\[
y = -3kx + 4k
\]
Comparing this with the given format \( y = \text{-----}x + \text{-------} \), we can assign:
- \( -3k \) as the coefficient of \( x \)
- \( 4k \) as the constant term
Thus, the filled-in version of the second equation can be:
\[
y = -3x + 4
\]
Here, if \( k = 1 \):
- The coefficient of \( x \) is \( -3 \) (leading to \( -3x \))
- The constant term is \( 4 \) (leading to \( +4\))
Therefore, one possible set of values can be:
\[
y = -3x + 4
\]
For example, if you let \( k = 2 \):
\[
y = -6x + 8
\]
Essentially, any \( k \neq 0 \) will work, provided you adjust both \( x \) and the constant term accordingly.
To summarize, the completed equation can be:
\[
y = -3x + 4 \quad \text{(for \( k = 1 \))}
\]
or, more generally:
\[
y = -3kx + 4k \quad \text{where \( k \) can be any non-zero constant.}
\]