Question
3x + 4y = 1 and one equals blank has an infinite number of solutions
Answers
GPT-4o mini
In order for a system of equations to have an infinite number of solutions, the equations must be dependent, which means that one equation can be derived from another.
The equation you've provided is \(3x + 4y = 1\). To create an equation that has an infinite number of solutions together with this one, we can multiply the entire equation by a nonzero constant.
For example, if we multiply the entire equation by 2, we get:
\[
2(3x + 4y) = 2(1)
\]
This simplifies to:
\[
6x + 8y = 2
\]
Now, the system of equations is:
1. \(3x + 4y = 1\)
2. \(6x + 8y = 2\)
Since the second equation is simply a multiple of the first, these two equations represent the same line on a graph, thereby resulting in an infinite number of solutions.
So, you can say: "one equals \(6x + 8y = 2\)" to have an infinite number of solutions along with \(3x + 4y = 1\).
The equation you've provided is \(3x + 4y = 1\). To create an equation that has an infinite number of solutions together with this one, we can multiply the entire equation by a nonzero constant.
For example, if we multiply the entire equation by 2, we get:
\[
2(3x + 4y) = 2(1)
\]
This simplifies to:
\[
6x + 8y = 2
\]
Now, the system of equations is:
1. \(3x + 4y = 1\)
2. \(6x + 8y = 2\)
Since the second equation is simply a multiple of the first, these two equations represent the same line on a graph, thereby resulting in an infinite number of solutions.
So, you can say: "one equals \(6x + 8y = 2\)" to have an infinite number of solutions along with \(3x + 4y = 1\).