Question

How do you solve `4x-9=6x+19`?

Answer 1

Method in details showing the basis of manipulation shortcuts.

`color(green)(x=-14)`

Explanation:

`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")`