How do you solve the following linear system: #3x + 5y = -1, 5x - y = 3#?

2 Answers
Jan 12, 2016

Answer:

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

Explanation:

Given
[1]#color(white)("XXX")3x+5y=-1#
[2]#color(white)("XXX")5x-y=3#

By substitution
Re-write [2] as
[3]#color(white)("XXX")y=5x-3#

Substitute #(5x-3)# for #y# in [1]
[4]#color(white)("XXX")3x+5(5x-3)=-1#

Simplify
[5]#color(white)("XXX")3x+25x-15=-1#

[6]#color(white)("XXX")28x=14#

[7]#color(white)("XXX")x=1/2#

Substitute #1/2# for #x# in [2]
[8]#color(white)("XXX")5(1/2)-y=3#

[9]#color(white)("XXX")y=5/2-3=-1/2#

Jan 12, 2016

Answer:

#(x,y)->(1/2,-1/2)#
#color(brown)("Finding x is explained in detail. Finding y is explained using shortcuts.")#

Explanation:

Given:

#color(brown)(3x+5y=-1)# ........................................(1)

#color(brown)(5x-y=3)#..................................................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("To find "x)#

From (2) we have #y=5x-3...........(2_a)#

Substitute #(2_a)#into (1) giving:

#3x+5(5x-3)=-1#

#3x+25x-15=-1#

#color(brown)(28x-15=-1)#
'-----------------------------------------------------------
Add #color(blue)(15)# to both sides giving:

#color(brown)(28x-15color(blue)(+15) =-1 color(blue)(+15))#

#28x=14#
'--------------------------------------------------------
Divide both sides by 28 which is the same as #color(blue)(xx1/28)#

#28/28 xx x= 14xx 1/28#

but #28/28=1" and " 14/28 = 1/2# giving:

#color(magenta)(x=1/2)#...........................................(3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

#color(blue)("To find y")# (Using shortcuts)

Substitute (3) into (1) or (2): Using (2) is easier!

#5x-y=3 -> 5(1/2)-y=3#
#color(magenta)(y=2 1/2 -3 =-1/2)#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Putting it all together")#

#(x,y)->(1/2,-1/2)#