A. −6x + 3y = 18 4x − 3y = 6
B 2x + 4y = 24 6x + 12y = 36
C. 3x − y = 14 −9x + 3y = −42
D. 5x + 2y = 13 −x + 4y = −6
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:
2x + 4y = 24
6x + 12y = 36
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:
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)