For each system please determine it has solutions?

Question

1)

-2x+ 5 y= 27
6x-15y= -81

A. No solutions
B. Unique solution: x=0, y=0
C. Unique solution: x=27, y=-81
D. Infinitely many solutions
E. Unique solution: x=-13.5, y=0
F. None of the above


2)

6x+3y=15
18x+9y=46

A. Unique solution: x=46, y=15
B. No solutions
C. Unique solution: x=0,y=0
D. Unique solution: x=15,y=46
E. Infinitely many solutions
F. None of the above

3)

-7x+10y=-102
8x+5y=18

A. Unique solution: x=-6,y=6
B. No solutions
C. Unique solution: x=0, y=0
D. Infinitely many solutions
E. Unique solution: x=6, y=-6
F. None of the above

4)

2x-8y=0
9x-8y=0

A. Unique solution: x=0,y=0
B. No solutions
C. Unique solution: x=8, y=2
D. Unique solution: x=-6, y=1
E. Infinitely many solutions
F. None of the above

Answer 1

-2x+ 5y= 27 ---> multiplying each term by -3 gives

6x-15y= -81

so D - every point on the line -2x + 5y = 27 is a solution





6x+3y=15 ----> multiplying each term by 3 gives

18x+9y=45

This is parallel to 18x + 9y = 46; parallel lines do not meet

so B, there are no solution



-7x+10y=-102

8x+5y=18 ---> multiplying each term by 2 and subtracting the other equation leaves

16x + 7x + 10y - 10y = 36 + 102

23x = 138

x = 6 ---> plug into one of the equations to find y

-7*6 + 10y = -102

10y = -60

y = -6

E - unique solution (6, -6)



2x-8y=0 ---> subtract this from the next equation

9x-8y=0

-------------

7x = 0

x = 0

y = 0

A

Answer 2

1)



-2x+ 5 y= 27

6x-15y= -81



-6x+15y=81

6x-15y=-81

=========

0=0 infinite sol

D. Infinitely many solutions





2)



6x+3y=15

18x+9y=46



18x+9y=45

18x+9y=46

===========

0=-1 no sol

B. No solutions



3)



-7x+10y=-102

8x+5y=18

-7x+10y=-102

-16x-10y=-36

===========

-23x=-138

x=6

8(6)+5y=18, 48-18=5y, y=6

F. None of the above



4)

2x-8y=0

9x-8y=0

=======

-7x=0

x=0

2(0)-8y=0

y=0

A. Unique solution: x=0,y=0

Answer 3

1) {x = 0, y = 27/5}