Question

Determine which system below will produce infinitely many solutions.

2x + 5y = 24
2x + 5y = 42

3x - 2y = 15
6x + 5y = 11

4x - 3y = 9
-8x + 6y = -18

5x - 3y = 16
-2x + 3y = -7

Answer 1

4x - 3y = 9

-8x + 6y = -18

Explanation:

Since, a system of equation,

`a_1x+b_1y = c_1` ,
`a_2x+b_2y=c_2`

Then the system has infinitely many solution if,

`(a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2)`

For the system of equations,

2x + 5y = 24

2x + 5y = 42

`(2)/(2)=(5)/(5)\\eq (24)/(42)`

⇒ This system does not have infinitely many solutions,

Now, For the system of equations,

3x - 2y = 15

6x + 5y = 11

`(3)/(6)\\eq (-2)/(5)\\eq (15)/(11)`

⇒ This system does not have infinitely many solutions,

For the system of equations,

4x - 3y = 9

-8x + 6y = -18

`(4)/(-8)=(-3)/(6)= (9)/(-18)=-2`

⇒ This system has infinitely many solutions,

For the system of equations,

5x - 3y = 16

-2x + 3y = -7

`(5)/(-2)\\eq (-3)/(3)\\eq (16)/(-7)`

⇒ This system does not have infinitely many solutions,