For the system of equations to have an infinite number of solutions, the second equation must be equivalent to the first one when rearranged.
The first equation is given as:
\[ 3x + 4y = 1. \]
To express \( y \) in terms of \( x \), we can solve for \( y \):
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Subtract \( 3x \) from both sides:
\[ 4y = 1 - 3x. \]
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Divide by 4:
\[ y = \frac{1 - 3x}{4}. \]
Now any multiple or transformation of this equation can also represent the same line. For instance, if we multiply the entire equation by \( 2 \), we get:
\[ y = \frac{2(1 - 3x)}{4} = \frac{1 - 3x}{2}. \]
Thus, a valid equation for \( y \) that would give the system infinite solutions is:
\[ y = \frac{1 - 3x}{4}. \]
To ensure it's clear:
Answer: \( y = \frac{1 - 3x}{4} \) (or any equivalent form such as \( y = \frac{1 - 3x}{2} \)).