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

#W/9=9#

Multiply both sides by #9#.

#W/cancel 9xxcancel 9=9xx9#

#W=81#

#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.

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#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:

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#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#
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#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#