The typical equation of a line is: y = mx + y₀ → where m: slope and where y₀: y-intercept
3x + 2y = 1 ← this is the line (ℓ₁)
2y = - 3x + 1
y = (- 3x + 1)/2
y = (3/2).x + (1/2)
7x + 5y = k ← this is the line (ℓ₂)
5y = - 7x + k
y = (- 7x + k)/5
y = - (7/5).x + (k/5)
Two lines are parallel if they have the same slope. It's not the case here because the slopes are different. So whatever the value of k, there is a unique solution.
As the slopes are different, the 2 lines are not parallel, so whatever the value of k, there is a unique solution.
Infinitely many solutions
The lines must be superposed, so they must have the same slope and they must have the same y-intercept.
First condition: the same y-intercept
1/2 = k/5
k = 5/2
…but even if the lines have the same y-intercept, as their slope are different, these 2 lines are not parallel, so they can be superposed. So, obtaining parallel lines is not possible, so obtaining infinitely many solutions is not possible.