Question

Determine which system below will produce infinitely many solutions.

-4x + 6y = 20
4x - 3y = -2

-2x + 4y = 18
4x - 8y = -36

3x - 2y = 5
-6x - 4y = -10

5x - 3y = 12
15x - 9y = 24

Answer 1

There are basically three posibilities

The lines cross in one point, so in that point both equations are true
The crossing point is the solution of the system of equations

When the lines are parallel, there are 2 options
If the lines are in fact the same line, all points on that line both equations are true for all x and y on the line, and there are infinitely many solution

Or the lines of the system are parallel and cross nowhere, then there is no solution.

Answer 2

-4x + 6y = 20
4x - 3y = -2
If you add the second eq. to the second, you get
3y = 18
so
y = 6
substitute y = 6 into both original equations gives
-4x + 36 = 20
subtract 36 from both sides
-4x = -16
so divide both sides by -4
x = 4

substitute the found solution into the second equation to see if the solution is valid for both equations
4*4 - 3*6 = -2, so the solution is valid and uniqe


4x - 3y = -2

Answer 3

for set 2, multiply the first eq. by -2
-2x + 4y = 18
4x - 8y = -36
then we get the following system
4x - 8y = -36
4x - 8y = -36
which is a whole line of solutions, thus there are infinitely many

Answer 4

Set 3
3x - 2y = 5
-6x - 4y = -10
multiply the first by 2 and the second by -1
6x - 4y = 10
6x + 4y = 10
substact the first eq. from the second
and we get
8y = 0
so
y = 0

Now substitute this in the two original equations
3x = 5
x = 5/3

and y = 0 and x = 5/3 into
-6x - 4y = -10
gives
-6*5/3 = -10
which is true, so there is only one uniqe solution

Answer 5

set 4
5x - 3y = 12
15x - 9y = 24
multiply the first equation by 3
15x - 9y = 36
15x - 9y = 24
now you can already see that the lines are parallel but not the same
if you substract the second from the first then you get
0 = 12

so there are no solutions (the lines are parallel and do not coincide)

Answer 6

In summary
Only set 3 has infinitely many solutions