For the system of equations to have an infinite number of solutions, the second equation must be a multiple of the first equation.
Starting with the equation \( 3x + 4y = 1 \), we can solve for \( y \):
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Rewrite the original equation:
\( 4y = 1 - 3x \) -
Divide everything by 4:
\( y = \frac{1}{4} - \frac{3}{4}x \)
Now, say we multiply the entire equation by any non-zero constant \( k \).
For example, let \( k = 2 \): \[ y = 2 \left(\frac{1}{4} - \frac{3}{4}x\right) = \frac{1}{2} - \frac{3}{2}x \]
The equation that results is: \[ y = -\frac{3}{2}x + \frac{1}{2} \]
Thus, completing the statement: The system of equations 3x + 4y = 1 and \( y = -\frac{3}{2}x + \frac{1}{2} \) has an infinite number of solutions.
You could also choose other multiples, but they all must reflect the linear relationship derived from the original equation.