For infinitely many solutions, we are looking for linearly dependent equations, which means that one equation is an exact multiple or sub-multiple of the other. Example: </span>2x + 4y = 24 6x + 12y = 36 <span>is a system that does NOT have a solution, because 6/2=3 for x, 12/4=3 for y, but 36/24=1.5. The two lines have the same slope (therefore parallel), but they have different y-intercepts. So the two lines will never meet, and therefore no solution.
or another example: </span> 3x -y = 14 -9x + 3y = -42 We see that -9/3=-3, 3/-1=-3, -42/3=-14, this system has coefficients all in the same ratio, meaning that the lines are coincident (and linearly dependent), therefore infinitely many solutions.
Still another example: −6x + 3y = 18 4x − 3y = 6 we see that 18/6=3, 3/(-3)=-1 , since the ratios are different, the two equations are not linearly dependent, and therefore the system has unique solution.
Last example: 5x + 2y = 13 −x + 4y = −6 Check the ratio of the coefficients: -1/5=-1/5 4/2=2 ..... we can stop here and conclude that there is a unique solution because the equations are not in the same ratio. (unique means that there is exactly one solution)
1/2 and 3/5 can be written as 20/40 and 24/40. Between them you can have 3 rational numbers as 21/40, 22/40, and 23/40, which may be written as 21/40, 11/20 and 23/40.
