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{align} 5x + (x-y)=21&\\

-\underline{\quad 5y- (x-y)=14}&\\

5x-5y+2(x-y)=\ 7&\\

\ \ \\

\Rightarrow 5(x-y)+2(x-y) =\ 7&\\

\Rightarrow x-y =\ 1&\end{align}$

Finding $x$ and $y$ first

Substitution

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

$6y-x=14$ becomes:

         $\begin{align}6(6x-21)-x&=14\\

\Rightarrow 36x-126-x&=14\\

\Rightarrow 35x&=140\\

\Rightarrow x&=4\end{align}$

Elimination

$6x-y=21$

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

Add:

$\begin{align}\ 6x\ -\ y\ =\ 21&\\

+\underline{\quad 36y-6x=\ 84\ }&\\

35y=105&\\

\Rightarrow y =\ \ 3\ \ &\end{align}$