Question

How do you solve for r in `4(r+3)=t`?

Answer 1

r=t-12/4

Explanation:

4[r+3]=t
4r+12=t
4r=t-12
r=t-12/4

Answer 2

`color(green)(" "r=t-3)`

Explanation:

Why the shortcut method works:

`color(blue)("Explaining from first principles about multiplying out a bracket")`

Consider the following:
`2a" is the same as "a+a`
`3a" is the same as "a+a+a`
`4a" is the same as "a+a+a+a`

So it follows that

`4(r+3)" is the same as " (r+3)+(r+3)+(r+3)+(r+3)`

From this you can see that there is 4 lots of r and 4 lots of 3's

So when multiplying out the bracket we have the equivalent of

`color(brown)(4color(blue)((r+3))" "->" "(4color(blue)(xxr))color(blue)(+)(4color(blue)(xx3))`

So `4(r+3) = 4r+12`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Solving your question")`

Given:`" "4(r+3)=t`

This is the same as:

`" "color(brown)(4r+12=t)`

We are told that we need to solve for r. That means we must end up with only one r in the equation and that it is to be on the left hand side of the equals sign. Everything else is to be on the other side.

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

Collect all the terms with r on the left of = and everything else on the right. So we need to move the 12 to the other side.

To remove 12 from the left we do the following

Subtract `color(blue)(12)` from both sides

`" "color(brown)(4r+12color(blue)( -12)=tcolor(blue)(-12))`

But `+12-12=0`

`" "4r+0=t-12`

`" "4r=t-12`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Step 2")`

Any number multiplied by 1 does not change its value. We need r on its own. So we change the 4 into 1

Divide both sides by `color(blue)(4)`

`" "(4r)/4=t/4-12/4`

This is the same as

`" "4/4xxr=t/4-12/4`

But `4/4 = 1` giving

`" "1xxr=t-12/4`

`" "r=t/4-12/4`

But `-12/4 = -3`

`" "r=t/4-3`

'///////////////////////////////////////////////////////////////////////
`color(blue)("The short cut rules that achieve the same thing are:")`

`color(brown)("To move something to the other side of the equals.")`

For add, move the term to the other side of the = and change the sign to subtract.

For subtract, move the term to the other side of the = and change the sign to add.

For multiply, move the term to the other side and divide by it.

For divide, move the term to the other side of the = and multiply by it.

Hope this helps in solving other question.