How do you solve the following system?: #-5x -2y =3 , 2x -y = -6#

1 Answer
Nov 12, 2015

Answer:

Shown two methods. The first in detail. Solved for #x# and left finding the value of #y# for the question proposer.

Explanation:

#color(brown)("You have 2 unknowns and 2 equations so it is solvable.")#

#color(brown)("Two ways of solving:")#

#color(blue)("Method 1")#

In detail: every step shown!!
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider #-5x-2y=3 # making #y# the dependant variable

#color(brown)("Showing the method for changing the variable for "5x-2y=3" only.")#

Add #color(green)(2y)# to both sides giving:
# (-5x-2y) color(green)(+2y) = (3)color(green)(+2y)#

#-5x = 3 +2y#

Subtract #color(green)(3)# from both sides giving:
#(-5x) color(green)(-3) =(2y+3)color(green)(-3)#

#-5x-3=2y#

Divide both sides by #color(green)(2)#
Note that divide by 2 is the same as multiply by #1/2#

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

#-5/2x -3/2 = y#
Write as #color(blue)(y=-5/2x-3/2 ....................................(1))#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(brown)("Changing the variable for "2x-y =-6)#

#color(blue)(y=2x+6........................................................(2))#

Subtract #color(green)((1))# from (2) giving:

#y - color(green)(y)=2x -color(green)((-5/2x)) +6 -color(green)((-3/2))#
#0=9/2 x + 15/2#
#9/2x = -15/2#

Multiply throughout by 2 giving:
#9x=15#

#color(blue)(x=15/9................................................(3))#

Next step I will leave for you to do!
Substitute (3) into either of (1) or (2) to find the value of y

#color(blue)("Method 2")#

#-5/2x-3/2 =y =2x+6#

#-5/2x-3/2 =2x+6#

Now solve for x and substitute in (1) or (2) to find y

#color(red)("Both methods are fast once you are")#

#color(red)("able to "underline("change the variable by sight")#