Algebraic differences

Answer: $1$

Finding $x-y$ directly

$6x-y=21$ can be written as $5x + (x-y) = 21$

$6y-x=14$ can be written as $5y - (x-y) = 14$

Subtract:

$$$\begin{array}{rl}5x+(x-y)=21& \\ -\underset{―}{5y-(x-y)=14}& \\ 5x-5y+2(x-y)=7& \\ \\ \Rightarrow 5(x-y)+2(x-y)=7& \\ \Rightarrow x-y=1& \end{array}$$$

Finding $x$ and $y$ first

Substitution

$6x-y=21\Rightarrow y=6x-21$

$6y-x=14$ becomes:

         $$$\begin{array}{rl}6(6x-21)-x& =14\\ \Rightarrow 36x-126-x& =14\\ \Rightarrow 35x& =140\\ \Rightarrow x& =4\end{array}$$$

Elimination

$6x-y=21$

$6y-x=14\Rightarrow 36y-6x=84$

Add:

$$$\begin{array}{rl}6x-y=21& \\ +\underset{―}{36y-6x=84}& \\ 35y=105& \\ \Rightarrow y=3& \end{array}$$$