For the system of equations to have an infinite number of solutions, the second equation must be a multiple or a rearrangement of the first equation.
Given the first equation:
\[ 3x + 4y = 1 \]
We can express \(y\) in terms of \(x\). Rearranging the equation gives:
\[ 4y = 1 - 3x \]
Dividing everything by 4:
\[ y = \frac{1}{4} - \frac{3}{4}x \]
Now, to provide a second equation that results in an infinite number of solutions, we can multiply the first equation by a non-zero constant or simply rearrange the same equation.
For example, we can multiply the entire equation by \(k\), where \(k\) is any non-zero constant. Let's take \(k = 1\) for simplicity.
Thus, the second equation could also be:
\[ y = \frac{1}{4} - \frac{3}{4}x \]
Or more generally:
\[ 3x + 4y = k(1) \] where \( k \) is any non-zero constant.
By using \(y = \frac{1}{4} - \frac{3}{4}x\) as the second equation, we create a system that has infinite solutions along the line represented by this equation.
Therefore, the statement can be completed as follows:
The system of equations \(3x+4y=1\) and \(y = \frac{1}{4} - \frac{3}{4}x\) has an infinite number of solutions.