Question

How do you solve `3+2g=5g-9`?

Answer 1

`g=4`

Explanation:

`3+2g=5g-9`
`3+9==5g-2g`
`12=3g`
`12/3=g`
`4=g`

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Check

`3+2(4)=5(4)-9`
`3+8=20-9`
`11=11`

Answer 2

`g=4`

Using first principles, which is where the shortcuts come from.

Explanation:

Given:`" "3+2g=5g-9`

`color(blue)("Method")`

Manipulate the expression so that you have only the terms with g on one side of the equals sign.

Manipulate the terms with g so that you only have one g on one side of the equals and everything else on the other side.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Isolating the terms with "g)`

Add `color(blue)(9)` to both sides

`color(brown)(3+2gcolor(blue)(+9)=5g-9color(blue)(+9))`

But `3+9=12" "` and `" "-9+9=0`

`12+2g=5g+0`

Subtract `2g` from both sides

`color(brown)(12+2gcolor(blue)(-2g)=5gcolor(blue)(-2g))`

But `2g-2g=0" "` and `" "5g-2g=3g`

,.......................................................................................
`color(blue)("Isolating "g)`

`12+0=3g`

Divide both sides by `color(blue)( 3)`

`color(brown)(12/(color(blue)(3)) =(3g)/(color(blue)(3))`

`color(brown)(12/(color(blue)(3)) =3/(color(blue)(3))xx g)`

But `3/3=1`

`12/3=1xxg`

`g=12/3`

`g=(12-:3)/(3-:3) = 4/1 = 4`