<s>[INST] Solving for unknown When solving for unknowns why does the symbols sometimes change from $a + $ to $a -$ or vice versa or stays the same?
For example (using solutions from text): 
$2x+6 = x+16\quad$ and the solution is $2x-x = 16-6\quad \Rightarrow \quad x=10$.
Or
$8x-12 = -3x+21\quad $Solution: $8x+3x = 21+12\quad \Rightarrow \quad
                 11x = 33\quad \Rightarrow \quad
                 x=3$
 [/INST] 
For example: $2x+6 = x+16$, and the solution is $2x-x = 16-6\Rightarrow x=10$.

Here, let's start by adding $-x$ to both sides of the equation $2x +6 = x+16$: $$-x + 2x+6 = -x + x + 16$$ $$\leftrightarrow x+6 = 16.$$
Next: Adding $-6\,$ to each side of the equation gives us:  $$x + 6 + (-6) = 16 + -6$$ $$\leftrightarrow x + 0 = 16 - 6 $$ $$\leftrightarrow x = 10$$

Solving for $x$ given $8x-12 = -3x+21$:

Add $3x$ to each side of the equation, AND add $12$ to both sides of the equation.
This gives: $$3x + 8x -12 + 12 = 3x + -3x + 21 + 12$$ $$\leftrightarrow 11x + 0 = 0 + 21 + 12$$ $$\leftrightarrow 11x =33$$  $$\leftrightarrow x = 3.$$ 
Note that in the last step, we divided both sides of the equation by $11$ (which is the same as multiplying both sides of the equation by $\frac{1}{11}$).
The objective when solving for one unknown, say $x$, is to get all multiples of $x$ on the left hand side, and all constants on the right hand side. Then simplify. As long as you are adding (or subtracting) the same value from each side of the equation, and as long as you multiply (or divide) both sides of an equation by the same value, the equality remains unchanged.
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