No graph, no problem

Question

Mai is given these two systems of linear equations to solve:

She analyzed them for a moment, and then—without graphing the equations—said, "I got it! One of the systems has no solution and the other has infinitely many solutions!" Mai is right!

Which system has no solution and which one has many solutions? Explain or show how you know (without graphing the equations). 

Answer 1

They look suspiciously similar, Mai can see that straight away that multiplying the top equation by 4 gives the same coefficients as equation 2.

5x + y = 13 becomes 20x + 4y = 52

subtract 20x + 4y = 64 to get 0x + 0y = -12 => 0 = -12 which is false, hence no solutions.

while 5x + y = 13 becomes 20x + 4y = 52

and 20x = 52 - 4y becomes 20x + 4y = 52

subtract to get 0x + 0y = 0 => 0 = 0 which is true, so infinite solutions.