Infinite solutions to a system of equations

When two equations turn out to be the same line, they share every point, which means infinitely many solutions.
The clue is that one equation is a multiple of the other.
Multiplying $x + y = 2$ by $2$ gives $2x + 2y = 4$, so those two are really one line.

On a graph, the two lines must have the same slope and the same $y$-intercept.
Same slope but different intercepts gives parallel lines and no solution.
Same slope and same intercept means the lines lie exactly on top of each other.

You can also check the numbers directly.
For $ax + by = c$ and $px + qy = r$, there are infinitely many solutions when $\frac{a}{p} = \frac{b}{q} = \frac{c}{r}$.
That means every part of one equation is the same multiple of the other.

When you try to solve such a system, the variables all cancel and you get a true statement like $0 = 0$.
That is the algebra telling you the two equations are the same line.
A false statement like $0 = 4$ would instead mean no solution.

How many solutions does the system $2x + y = 5$ and $4x + 2y = 10$ have?
The second equation is the first multiplied by $2$, so they are the same line.
So the system has infinitely many solutions.

A system has infinitely many solutions, and one equation is $4x - 6y = 12$. Which could be the other: $2x - 3y = 6$ or $2x - 3y = 10$?
Dividing $4x - 6y = 12$ by $2$ gives $2x - 3y = 6$, the same line.
So $2x - 3y = 6$ works; $2x - 3y = 10$ is a parallel line with no solution.

For $ax - 3a + 4 = 6x + 2 + b$ to have infinitely many solutions, what must be true?
Both sides must be identical, so the $x$ coefficients match and the constants match.
That gives $a = 6$ and $-3a + 4 = 2 + b$.