How do you solve the following system?: #2x -5y =9 , 5x -3y = 2#

1 Answer
Nov 18, 2015

Answer:

#y=-41/19color(white)(xxxx)x=-17/19#

Keeping in fraction form improves precision

Explanation:

Given:
#2x-5y=9..................................(1)#
#5x-3y=2..................................(2)#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Consider Equation (1)")#

Add #5y# to both sides
#2x=9+5y#

Subtract 9 from both sides.
#2x-9 =5y#

Divide both sides by 5
#2/5 x-9/5 =y#

Write in conventional form
#color(green)(y=2/5x-9/5....................................(1_a))#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Consider Equation (2)")#

Add #3y# to both sides
#5x=2+3y#

Subtract 2 from both sides
#5x-2=3y#

Divide both sides by 3
#5/3 x-2/3 = y#

Write in conventional form
#color(green)(y=5/3x-2/3....................................(2_a))#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
To end up with just 1 variable thus solvable.

#color(green)("Equation "(2_a))color(red)( =y= )color(blue)("Equation "(1_a)")#

#color(green)(2/5x-9/5) color(red)(=) color(blue)(5/3x-2/3)#

Collecting like terms
#2/5x -5/3x =9/5 -2/3#

#(6 -25)/15 x = (27 -10)/15#

#-19/15 x =17/15#

#-19x=17#

#color(red)(x=-17/19...........................(3))#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute for #x# in Equation (1)

#2x-5y=9..................................(1)#

#2color(red)((-17/19)) -5y=9#

#5y= 2(-17/19) -9#

#y= 2/5(-17/19)-9/5#

#color(red)(y=-2 3/19=-41/19...............(4))#