If x<0and y<0 x+y+x÷y=1÷2 and (x+y)x÷y=-1÷2 then find x and y?

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Answer 1 · 2018-06-07T10:01:21

# x = -1/4 # and #y=-1/4#

We have:

# x + y +x/y=1/2# ..... [A]
#(x+y)x/y=-1/2# ..... [B]

with #x,y lt 0#

From [A] we have:

# x+y = 1/2-x/y #

Substituting into [B] we get:

#(1/2-x/y )x/y=-1/2#

Substitute #u=x/y#, noting that #u gt 0 \ because x,y lt 0#:

#(1/2-u )u=-1/2#

#:. u - 2u^2 = -1#

#:. 2u^2-u-1 = 0#

#:. (2u+1)(u-1) = 0#

Leading to two possible solutions:

# u = 1# or # u = -1/2#

We discard the negative solution as #x,y gt =>u lt o# leaving us with #u=1 => x/y=1 => x=y#, so substituting #=x# in [A], we get:

# x + x +x/x=1/2#

# :. 2x+1 = 1/2 #

# :. x = -1/4 # and #y=-1/4#