For the system of equations to have an infinite number of solutions, the second equation must be a multiple of the first equation. This means it must represent the same line as the first equation.
Starting with the equation \(3x + 4y = 1\), we can solve it for \(y\):
\[ 4y = 1 - 3x \quad \text{(subtracting } 3x \text{ from both sides)} \]
\[ y = \frac{1 - 3x}{4} \quad \text{(dividing both sides by 4)} \]
Now, we can write the second equation \(y = ?\) as any multiple of this equation. For example, if we multiply the entire equation by 2, we get:
\[ y = \frac{2(1 - 3x)}{4} = \frac{2 - 6x}{4} = \frac{1 - 3x}{2} \]
Thus, the completed statement could be:
The system of equations \(3x + 4y = 1\) and \(y = \frac{1 - 3x}{4}\) has an infinite number of solutions.
Or you could express it with any other scalar multiple:
The system of equations \(3x + 4y = 1\) and \(y = \frac{1 - 3x}{k}\) where \(k\) is any non-zero constant, has an infinite number of solutions.