#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!!
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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))#
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#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")#
