Question

How do you solve `W/9 = 9`?

Answer 1

`W=81`

Explanation:

`W/9=9`

Multiply both sides by `9`.

`W/cancel 9xxcancel 9=9xx9`

`W=81`

Answer 2

`color(green)("An expansion on the solution method to deepen understanding of")` `color(green)("what is going on.")`

Explanation:

`color(blue)("Short cut method")`
For add or subtract move to the other side of the equals and do the opposite: change its sign.

For divide or multiply move it to the other side of the equals and do the opposite: if originally divide then you multiply when on the other side. If originally multiply then you divide when on the other side.

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(red)("Why the shortcuts work")`

The key to manipulation of equations and moving something to the other side of the equals sign:

`color(green)("For multiply or divide you change it to 1")`
`color(purple)("For add or subtract you change it to 0.")`

Let me demonstrate:

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Example 1")`

Suppose we had `color(brown)(x+2=3)` and we whish to find the value of `x`

To end up with `x` on its own we need to remove the `+2`

Subtract `color(blue)(2)` from both sides:

`color(brown)((x+2)color(blue)(-2) = ( 3 )color(blue)(-2)`

`x+(2-2) =3-2`

`x+0=1`

`x=1`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Example 2")`

Suppose we had `color(brown)(2x=3)`

Divide both sides by `color(blue)(2)`
But divide by 2 is the same as multiply by `color(green)(1/2)`

`color(brown)( 2x color(green)(xx1/2) =3color(green)(xx1/2))`

`color(brown)( (2color(green)(xx1))/(color(green)(2)) xx x= (3color(green)(xx1))/(color(green)(2))`

`color(brown)(2/(color(green)(2)) xx x=3/(color(green)(2))`

But `2/2 = 1` so we have:

`1xx x = 3/2`

`x=3/2`
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`color(blue)("For your question:")`

Multiply both sides by 9 giving:

`W/9xx 9 =9xx9`

`Wxx 9/9=81`

`W=81`