Question

Determine which system below will produce infinitely many solutions.

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

i really need help

Answer 1

The answer is the option C

`3x-y=14`

`-9x+3y=-42`

Explanation:

case A) we have

`-6x+3y=18` ------> equation A

`4x-3y=6` ------> equation B

Adds equation A and equation B

`-6x+3y=18 \\4x-3y=6\\--------- \\-6x+4x=18+6 \\-2x=24 \\ x=-12`

Find the value of y

substitute in the equation A the value of x

`-6(-12)+3y=18`

`3y=18-72`

`y=-18`

therefore

the system case A) has one solution

case B) we have

`2x+4y=24` ------> equation A

`6x+12y=36` ------> equation B

Multiply by
`3` equation A

`3(2x+4y)=3*24`

`6x+12y=72`

so

Equation A and equation B are parallel lines

therefore

the system case B) has no solution

case C) we have

`3x-y=14` ------> equation A

`-9x+3y=-42` ------> equation B

Multiply by
`3` equation A

`3(3x-y)=3*14` ------->
`-9x+3y=-42`

Equation A and equation B are the same line

therefore

the system case C) has infinitely solutions

case D) we have

`5x+2y=13` ------> equation A

`-x+4y=-6` ------> equation B

Multiply by
`5` equation B

`5(-x+4y)=-6*5`

`-5x+20y=-30` -------> equation C

Adds equation A and equation C

`5x+2y=13\\-5x+20y=-30\\------------\\2y+20y=13-30 \\22y=-17 \\y=-0.77`

Substitute the value of y in the equation B

`-x+4(-0.77)=-6`

`-x=-6+3.09`

`x=2.91`

the system case D) has one solution

Answer 2

Correct answer is C.

System of two equivalent equations will produce infinitely many solutions.

`3x-y=14`
Multiply by (-3)

`(-3)3x-(-3)y=(-3)14 \\-9x+3y=-42`
So it is equivalent to the second equation.