Question
Cruz road assistance movie of equations with an infinite number solutions one of his equations is 6X + 3y = 9 what could be his second equation right to you equation and slope intercept form the second equation could be y equals
Answers
GPT-4o mini
To find a second equation that has an infinite number of solutions with the first equation \(6x + 3y = 9\), the second equation must be a multiple of the first equation. This means it should have the same slope and thus be equivalent after simplification.
First, let's rewrite the first equation in slope-intercept form (i.e., \(y = mx + b\)):
1. Start with the original equation:
\[
6x + 3y = 9
\]
2. Subtract \(6x\) from both sides:
\[
3y = -6x + 9
\]
3. Divide every term by \(3\) to solve for \(y\):
\[
y = -2x + 3
\]
Now we have the slope-intercept form of the first equation: \(y = -2x + 3\).
To create a second equation with an infinite number of solutions, you can multiply the entire first equation by any non-zero constant. For example, let's multiply the first equation by \(2\):
1. Start with the original equation:
\[
6x + 3y = 9
\]
2. Multiply through by \(2\):
\[
12x + 6y = 18
\]
Now, we can also express this new equation in slope-intercept form by following the same procedure as before. Let's solve for \(y\):
1. Subtract \(12x\) from both sides:
\[
6y = -12x + 18
\]
2. Divide every term by \(6\):
\[
y = -2x + 3
\]
So the second equation, which has an infinite number of solutions with the first equation, can be written as:
\[
y = -2x + 3
\]
In conclusion, both equations \(6x + 3y = 9\) and \(12x + 6y = 18\) (or \(y = -2x + 3\)) represent the same line, leading to an infinite number of solutions.
First, let's rewrite the first equation in slope-intercept form (i.e., \(y = mx + b\)):
1. Start with the original equation:
\[
6x + 3y = 9
\]
2. Subtract \(6x\) from both sides:
\[
3y = -6x + 9
\]
3. Divide every term by \(3\) to solve for \(y\):
\[
y = -2x + 3
\]
Now we have the slope-intercept form of the first equation: \(y = -2x + 3\).
To create a second equation with an infinite number of solutions, you can multiply the entire first equation by any non-zero constant. For example, let's multiply the first equation by \(2\):
1. Start with the original equation:
\[
6x + 3y = 9
\]
2. Multiply through by \(2\):
\[
12x + 6y = 18
\]
Now, we can also express this new equation in slope-intercept form by following the same procedure as before. Let's solve for \(y\):
1. Subtract \(12x\) from both sides:
\[
6y = -12x + 18
\]
2. Divide every term by \(6\):
\[
y = -2x + 3
\]
So the second equation, which has an infinite number of solutions with the first equation, can be written as:
\[
y = -2x + 3
\]
In conclusion, both equations \(6x + 3y = 9\) and \(12x + 6y = 18\) (or \(y = -2x + 3\)) represent the same line, leading to an infinite number of solutions.