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 that both equations represent the same line.
Starting with the first equation:
\[ 3x + 4y = 1 \]
We can express \(y\) in terms of \(x\):
\[ 4y = 1 - 3x \] \[ y = \frac{1 - 3x}{4} \]
Now, we can express this in the form of \(y = mx + b\):
\[ y = -\frac{3}{4}x + \frac{1}{4} \]
Now to generate an equation that has the same slope and the same intercept, we can multiply the entire equation by any non-zero constant \(k\):
Let's choose \(k = 2\):
\[ 2(3x + 4y) = 2(1) \implies 6x + 8y = 2 \]
Thus, we can write:
\[ y = \frac{1}{4} - \frac{3}{4}x \]
So the equation can be presented in the form \(y = -\frac{3}{4}x + \frac{1}{4}\).
The complete statement can be:
The system of equations \(3x + 4y = 1\) and \(y = -\frac{3}{4}x + \frac{1}{4}\) has an infinite number of solutions.
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