Question
Fill in the blanks so that the following system of linear equations has infinite solutions.
y= -8x+8
y= -8x+8
Answers
GPT-4o mini
To ensure that the system of linear equations has infinite solutions, the second equation must be a multiple of the first equation. The first equation you provided is:
\[
y = -8x + 8
\]
To create a second equation that has infinite solutions with the first one, we can multiply both sides of the first equation by any non-zero constant \( k \).
Let's choose \( k = 2 \) as an example:
\[
2y = 2(-8x + 8)
\]
\[
2y = -16x + 16
\]
Now, if we rewrite this, we have:
\[
y = -8x + 8
\]
So our second equation, which has infinite solutions with the first, can be:
\[
2y = -16x + 16
\]
Thus, our system of linear equations is:
1. \( y = -8x + 8 \)
2. \( 2y = -16x + 16 \)
Both equations describe the same line and therefore have infinite solutions. You can choose any non-zero constant \( k \) to generate a valid second equation.
\[
y = -8x + 8
\]
To create a second equation that has infinite solutions with the first one, we can multiply both sides of the first equation by any non-zero constant \( k \).
Let's choose \( k = 2 \) as an example:
\[
2y = 2(-8x + 8)
\]
\[
2y = -16x + 16
\]
Now, if we rewrite this, we have:
\[
y = -8x + 8
\]
So our second equation, which has infinite solutions with the first, can be:
\[
2y = -16x + 16
\]
Thus, our system of linear equations is:
1. \( y = -8x + 8 \)
2. \( 2y = -16x + 16 \)
Both equations describe the same line and therefore have infinite solutions. You can choose any non-zero constant \( k \) to generate a valid second equation.