To create a system of equations that would make the statement true, we need to express \( y \) in terms of \( x \) based on the equation \( 3x + 4y = 1 \) and then derive another equation that is equivalent to it.
First, let's express \( y \) in terms of \( x \) from the given equation \( 3x + 4y = 1 \):
- Isolate \( 4y \): \[ 4y = 1 - 3x \]
- Solve for \( y \): \[ y = \frac{1 - 3x}{4} \]
Now, any equation that is a multiple of this equation will also represent the same line and therefore will have an infinite number of solutions when combined with \( 3x + 4y = 1 \).
One simple form of \( y \) that would work is a scalar multiple of the previous equation. For example, if we multiply the entire equation for \( y \) by 1, we still have:
\[ y = \frac{1 - 3x}{4} \]
Or, alternatively, we can express \( y \) as follows:
\[ y = -\frac{3}{4}x + \frac{1}{4} \]
Thus, the completed statement could be:
The system of equations \( 3x + 4y = 1 \) and \( y = -\frac{3}{4}x + \frac{1}{4} \) has an infinite number of solutions.