#color(green)(Given: color(white)(xx) 4x-9=6x+19)#
Objective: Have a single #x# on one side of the = and everything else on the other.
As #6x# is bigger than #4x# I am choosing to move the left #4x# to the right.
#color(blue)(Step 1)#
Subtract #color(blue)(4x)# from both sides; this will bring all the x-terms together
#color(brown)((4x-9) color(blue)(-4x) = color(brown)((6x+19) color(blue)(-4x)#
The purpose of the brackets is to show you what is being changed. They serve no other purpose than that or of grouping to make things clearer.
#color(brown)((4xcolor(blue)(-4x)) -9 =(6xcolor(blue)(-4x))+19)#
#0 -9 =2x+19#
#-9=2x+19#
~~~~~~~~~~This process explains the short cut ~~~~~~~~~~~~~~~~
By subtracting #4x# from both sides I have changed the one on the left to the value of 0. The consequence of this is that there is now a #4x# on the other side of the = but its sign has changed.
For addition and subtraction
#color(brown)("The shortcut is:" ) color(white)(x)color(blue)("move it to the other side of = and change its sign from + to -")#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)(Step 2)#
Subtract #color(blue)(19)# from both sides; this will isolate the x-terms.
#color(brown)((-9 )color(blue)( -19) = (2x+19))color(blue)(-19)#
#-28 = 2x +0#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)(Step 3)#
-
Divide both sides by 2 ( #divide 2 " is the same as" times 1/2#)
#color(brown)((-28))/(color(blue)(2)) = color(brown)((2x))/(color(blue)(2))#
#-14 = 2/2 x#
But #2/2 =1# giving
#-14=x#
Convention is that the #x# be written on the left so we have
#color(green)(x=-14)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For multiplication and division
#color(brown)("The shortcut is:" ) color(white)(x)color(blue)("move it to the other side of = and multiply by its inverse")#
