To determine which equation could be the other in a system that has an infinite number of solutions, we need to understand the criteria for such a system.
For a system of linear equations to have an infinite number of solutions, both equations must represent the same line. This means that one equation must be a scalar multiple of the other.
Given one of the equations in the system is:
[tex]\[ y = 3x - 1 \][/tex]
Let's analyze the options to see which could be the other equation in the system:
1. [tex]\( y = 3x + 2 \)[/tex]
- This equation is not a multiple of [tex]\( y = 3x - 1 \)[/tex] because the constant term is different. Thus, it cannot be the other equation.
2. [tex]\( 3x - y = 2 \)[/tex]
- Let's rearrange this in the slope-intercept form:
[tex]\[ y = 3x - 2 \][/tex]
- This equation is not a multiple of [tex]\( y = 3x - 1 \)[/tex] again because the constant term is different. Thus, it cannot be the other equation.
3. [tex]\( 3x - y = 1 \)[/tex]
- Let's rearrange this in the slope-intercept form:
[tex]\[ y = 3x - 1 \][/tex]
- This equation is exactly the same as [tex]\( y = 3x - 1 \)[/tex]. Hence, this equation could be the other in the system, making the system have an infinite number of solutions.
4. [tex]\( 3x + y = 1 \)[/tex]
- Let's rearrange this in the slope-intercept form:
[tex]\[ y = -3x + 1 \][/tex]
- This equation is not a multiple of [tex]\( y = 3x - 1 \)[/tex] because the slope is different. Thus, it cannot be the other equation.
Based on this analysis, the other equation that could be in the system providing an infinite number of solutions is:
[tex]\[ 3x - y = 1 \][/tex]
The correct option is:
[tex]\[ \boxed{3} \][/tex]
