Step 1: Use a concrete pair of equations instead of the general ratio rule.
Rather than quoting the $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ condition, let's build an actual pair of coincident lines and see directly how many points they share.
Step 2: Write two equations that describe the same line.
Take the line $x + y = 2$. Now multiply this whole equation by 2:
\[ 2x + 2y = 4 \]
This is a second equation, but it describes exactly the same set of points as $x + y = 2$, because dividing it by 2 gives back the first equation. When two equations describe the same line, their graphs coincide completely, they lie on top of each other.
Step 3: Count how many points these two lines share.
Pick any point on the line $x + y = 2$, for example $(0, 2)$, $(1, 1)$, or $(2, 0)$. Check each one in the second equation $2x + 2y = 4$:
For $(0, 2)$: $2(0) + 2(2) = 4$, true.
For $(1, 1)$: $2(1) + 2(1) = 4$, true.
For $(2, 0)$: $2(2) + 2(0) = 4$, true.
Every single point that lies on the first line also lies on the second line, because they are literally the same line drawn twice. Since a line contains infinitely many points, this means the pair of equations shares infinitely many common points.
Step 4: Connect this back to the general case.
A solution of a pair of linear equations is just a common point of the two lines. Two coincident lines, whichever numbers you plug in, always behave exactly like the example above, they are the same line counted twice, so they always share every one of their infinitely many points.
Step 5: Final answer.
A pair of equations represented by coincident lines has an infinite number of solutions, which is option (D).
\[ \boxed{\text{Infinitely many solutions}} \]