How to Solve bu using elimination method ?

# 5/(x+y) + 2/(x-y) = 3#
# 20/(x+y) - 3/(x-y) = 1#

3 Answers
Sonnhard
Jun 4, 2018

#x=-11/2,y=-9/2#

Explanation:

Substituting
#a=1/(x+y),b=1/(x-y)#
so we have the System
#5a+2b=3#
#20a-3b=1#
Multiplying the first equation by #-4# and adding to the second we get
#b=-1#
so
#a=-1/10#
Then we have
#1=-x+y#
#-10=x+y#
from here we get
#x=-11/2#
#y=-9/2#

Jumbotron
Jun 4, 2018

#x = 3 and y = 2#

Explanation:

#5/(x + y) + 2/(x - y)=3#

#20/(x + y)-3/(x - y)= 1#

Let;

#1/(x + y) = a and 1/(x - y) = b#

Therefore;

#5/(x + y) + 2/(x - y)=3#

#5(1/(x + y)) + 2(1/(x - y))=3#

#5a + 2b = 3 - - - eqn1#

Similarly..

#20/(x + y)-3/(x - y)= 1#

#20(1/(x + y))-3(1/(x - y)) = 1#

#20a - 3b = 1 - - - eqn2#

Using Elimination Method!

#5a + 2b = 3 - - - eqn1#

#20a - 3b = 1 - - - eqn2#

Multiply #eqn1# by #3# and #eqn2# by #2#

#3(5a + 2b = 3)#

#2(20a - 3b = 1)#

#15a + 6b = 9 - - - eqn3#

#40a - 6b = 2- - - eqn4#

Adding both #eqn3 and eqn4# together..

#(15a + 40a) + (6b + (-6b))= 9 + 2#

#55a + 6b - 6b = 11#

#55a = 11#

#a = 11/55#

#a = 1/5#

Substituting the value of #a# into #eqn1#

#5a + 2b = 3 - - - eqn1#

#5(1/5) + 2b = 3#

#cancel5(1/cancel5) + 2b = 3#

#1 + 2b= 3#

#2b = 3 - 1#

#2b = 2#

#b = 2/2#

#b = 1#

But;

#1/(x + y) = a and 1/(x - y) = b#

#a = 1/(x + y)#

#1/5 = 1/(x + y)#

Cross multiplying;

#1(x + y) = 1(5)#

#x + y = 5 - - - eqn5#

Similarly..

#b = 1/(x - y)#

#1 = 1/(x- y)#

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

Cross multiplying;

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

#x - y = 1 - - - eqn6#

Solving simultaenously again..

#x + y = 5 - - - eqn5#

#x - y = 1 - - - eqn6#

Using Elimination Method!

Adding #eqn5 and eqn6# together;

#(x + x) + (y +(-y)) = 5 + 1#

#2x + y - y = 6#

#2x = 6#

#x = 6/2#

#x = 3#

Substituting the value of #x# into #eqn6#

#x + y = 5 - - - eqn5#

#3 + y = 5#

Collecting like terms;

#y = 5 - 3#

#y = 2#

Hence;

#x = 3 and y = 2#

Barney V.
Jun 4, 2018

#x=3,y=2#

Explanation:

#5/(x+y)+2/(x-y)=3#
#20/(x+y)-3/(x-y)=1#

Let x+y be a and x-y be b

#:.5/a+2/b=3#----------(1)

#:.20/a-3/b=1-----------(2)#

#:.(1) xx 4#

#:.20/a+8/b=12#------(3)

#:.(2)-(3)#

#:.-11/b=-11/1#

#:.b/-11=1/-11#

multiply both sides by-11

#:.b=-11/-11#

#:.b=1#

substitute b=1 in (1)

#:.5/a+2=3#

#:.5/a=3-2#

#:.5/a=1#

#:.a/5=1#

multiply both sides by 5

#:.a=5#

#:.a=x+y=5#-----(4)

#:.b=x-y=1#------(5)

#:.(4)+(5)#

#:.2x=6#

#:.x=6/2#

#:.x=3#

#:.3+y=5#

#:.y=5-3# in (4)

#:.y=2#

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