# How do you solve -4r - 11 = 4r + 21?

Oct 17, 2016

$r \text{ "=" } - 4$

#### Explanation:

The objective is to have just have one $r$ and for it to be on one side of the = and everything else on the other.

I am going to use the first principles from which the short cuts are derived.

$\textcolor{b r o w n}{\text{For add or subtract change the value into 0 and it ends up on the other side of =}}$
$\textcolor{b r o w n}{\text{For divide or multiply change the value to 1 and it ends up on the other side of =}}$

You will see what I mean.
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$\text{Given: } - 4 r - 11 = 4 r + 21$

As the right hand side (RHS) $4 r$ is positive we change the left hand side (LHS) one to 0

$\textcolor{g r e e n}{\text{Add "color(blue)(4r)" to both sides}}$

color(brown)(-4rcolor(blue)(+4r)-11" "=" "4rcolor(blue)(+4r)+21

$\text{ "0-11" "=" } 4 r \textcolor{b l u e}{+ 4 r} + 21$

$\text{ "-11" "=" } 8 r + 21$
............................................................................................................

$\textcolor{g r e e n}{\text{Subtract "color(blue)(21)" from both sides}}$

$\textcolor{b r o w n}{\text{ "-11color(blue)(-21)" "=" } 8 r + 21 \textcolor{b l u e}{- 21}}$

$\text{ "-11-21" "=" } 8 r + 0$

$\text{ "8r" "=" } - 32$
.........................................................................................................

$\textcolor{g r e e n}{\text{Divide both sides by } \textcolor{b l u e}{8}}$

color(brown)(" "8/(color(blue)(8))r" "=" "-32/(color(blue)(8))

But $\frac{8}{8} = 1$ giving

$\text{ "r" "=" } - 4$

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Shortcut methods:

For add or subtract move the other side of = and reverse its sign

For multiply or divide move the other side of = and reverse its sine (action). So add becomes subtract and multiply becomes divide and so on

Examples:
" "color(brown)(2x=3" "->" " x=3/2)" "color(blue)(x/2=3" "->" "x=2xx3)

$\textcolor{b r o w n}{2 + x = 3 \text{ "->" " x=3-2)color(blue)(" "x-2=3" "->" } x = 3 + 2}$