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

Mar 17, 2016

r=t-12/4

#### Explanation:

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

Mar 17, 2016

$\textcolor{g r e e n}{\text{ } r = t - 3}$

#### Explanation:

Why the shortcut method works:

$\textcolor{b l u e}{\text{Explaining from first principles about multiplying out a bracket}}$

Consider the following:
$2 a \text{ is the same as } a + a$
$3 a \text{ is the same as } a + a + a$
$4 a \text{ is the same as } a + a + a + a$

So it follows that

$4 \left(r + 3\right) \text{ is the same as } \left(r + 3\right) + \left(r + 3\right) + \left(r + 3\right) + \left(r + 3\right)$

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 $4 \left(r + 3\right) = 4 r + 12$
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$\textcolor{b l u e}{\text{Solving your question}}$

Given:$\text{ } 4 \left(r + 3\right) = t$

This is the same as:

$\text{ } \textcolor{b r o w n}{4 r + 12 = t}$

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.

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$\textcolor{b l u e}{\text{Step 1}}$

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 $\textcolor{b l u e}{12}$ from both sides

$\text{ } \textcolor{b r o w n}{4 r + 12 \textcolor{b l u e}{- 12} = t \textcolor{b l u e}{- 12}}$

But $+ 12 - 12 = 0$

$\text{ } 4 r + 0 = t - 12$

$\text{ } 4 r = t - 12$

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$\textcolor{b l u e}{\text{Step 2}}$

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 $\textcolor{b l u e}{4}$

$\text{ } \frac{4 r}{4} = \frac{t}{4} - \frac{12}{4}$

This is the same as

$\text{ } \frac{4}{4} \times r = \frac{t}{4} - \frac{12}{4}$

But $\frac{4}{4} = 1$ giving

$\text{ } 1 \times r = t - \frac{12}{4}$

$\text{ } r = \frac{t}{4} - \frac{12}{4}$

But $- \frac{12}{4} = - 3$

$\text{ } r = \frac{t}{4} - 3$

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$\textcolor{b l u e}{\text{The short cut rules that achieve the same thing are:}}$

$\textcolor{b r o w n}{\text{To move something to the other side of the equals.}}$

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.