For the system of equations to have an infinite number of solutions, the second equation must be a multiple of the first equation or can be expressed in the same form.
Given the first equation:
\[ 3x + 4y = 1 \]
Let's express \( y \) in terms of \( x \) for this equation. To do this, we can solve for \( y \):
- Rearranging the equation: \[ 4y = 1 - 3x \]
- Dividing by 4: \[ y = \frac{1 - 3x}{4} \]
Now, to create a second equation that will give us an infinite number of solutions, we could multiply the entire first equation by a non-zero constant \( k \). For example, if we let \( k = 1 \), we would have:
\[ y = \frac{1 - 3x}{4} \Rightarrow \text{(as our second equation)} \]
Alternatively, we could also write it as:
\[ y = \frac{1}{4} - \frac{3}{4}x \]
Thus, the complete statement is:
The system of equations \( 3x + 4y = 1 \) and \( y = \frac{1 - 3x}{4} \) has an infinite number of solutions.