34.6k views
1 vote
Determine which system below will produce infinitely many solutions.

−6x + 3y = 18 4x − 3y = 6 2x + 4y = 24 6x + 12y = 36 3x − y = 14 −9x + 3y = −42 5x + 2y = 13 −x + 4y = −6

User Huiyan Wan
by
8.2k points

1 Answer

1 vote
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)

User Ditto
by
8.3k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories