# U can help me?solve the question with properties of equality

## properties of equality 5n + 16 = -4 -5n

Dec 2, 2017

$n = - 2$

#### Explanation:

The properties of equality basically state that when an equality is given, the left hand side is always equal to the right hand side.

Solving for 'n':

$5 n + 16 = - 4 - 5 n$

Put everything with 'n' on one side and every other number on the other side:

$5 n + 5 n = - 4 - 16$

Simplifying:

$10 n = - 20$

Dividing both sides by 10 to isolate 'n':

$\therefore n = - 2$

Dec 7, 2017

$n = - 2.0$

A lot of detail is given so that you can see where everything comes from. Once you get used to these the calculation becomes quite fast.

#### Explanation:

$\textcolor{b l u e}{\text{The teaching bit}}$

People generally show the shortcut approaches. These are basically remembering the consequences of using first principles.

First principles basis is:

The meaning of the equals sign is 'sacrosanct'. It is absolute and must always be complied with.

No mater what form the left and right side are their true values must be the same. In simple terms using an example: $2 = 2$

So if you add or subtract from one side you do same to the other.

If you multiply EVERYTHING on one side you do the same to the EVERYTHING on the other side and so on.

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

$\textcolor{b r o w n}{\text{For multiply or divide change it to 1}}$
$\textcolor{b r o w n}{\text{For add or subtract change it to 0}}$
$\textcolor{b r o w n}{\text{You will see what I mean as I am going to use these. }}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Answering the question using first principles}}$

Given: $\textcolor{w h i t e}{\text{d}} 5 n + 16 = - 4 - 5 n$

$\textcolor{b r o w n}{\text{Moving the "-5n" on the right to the left}}$

Add $\textcolor{red}{5 n}$ to both sides

$\textcolor{g r e e n}{5 n + 16 = - 4 - 5 n \textcolor{w h i t e}{\text{ddd")->color(white)("dd}} 5 n \textcolor{red}{+ 5 n} + 16 = - 4 - 5 n \textcolor{red}{+ 5 n}}$

color(white)("ddddddddddddddddddd")->color(white)("dddd")color(green)(10n color(white)("d")+16=-4color(white)("dd")+0)

$\textcolor{b r o w n}{\text{Moving the "16" on the left to the right}}$

Subtract $\textcolor{red}{16}$ from both sides

color(white)("ddddddddddddddddddd")->color(white)("d")color(green)(10n color(white)("d")+16color(red)(-16)=-4color(white)("dd")-16)

color(white)("ddddddddddddddddddd")->color(white)("d")color(green)(10n color(white)("ddd")+0color(white)("dd.")=color(white)("dd")-20)

$\textcolor{b r o w n}{\text{Moving the "10" from "10n" on the left to the right}}$

Divide both sides by $\textcolor{red}{10}$

color(white)("ddddddddddddddddddd")->color(white)("d")color(green)(10/color(red)(10)n color(white)("ddd")=color(white)("dd")-20/color(red)(10))

$\textcolor{w h i t e}{\text{ddddddddddddddddddd")->color(white)("d")color(green)(1xxn color(white)("ddd")=color(white)("dd}} - 2.0$

But $1 \times n$ is the same as just $n$ giving:

$n = - 2.0$