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")
