Question

What is the solution this system of equations? 2/3x+y=6 and -2/3x-y=2

Answer 1

$\bf \begin{cases} \cfrac{2}{3}x+y=6\\[0.8em] -\cfrac{2}{3}x-y=2 \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \stackrel{\textit{using the first equation}}{\cfrac{2}{3}x+y=6}\implies y=6-\cfrac{2x}{3} \\\\[-0.35em] ~\dotfill$

$\bf \stackrel{\textit{using the found \underline{y} in the second equation}}{-\cfrac{2x}{3}-\left( 6-\cfrac{2x}{3} \right)=2}\implies -\cfrac{2x}{3}-6+\cfrac{2x}{3}=2 \\\\\\ \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{3}}{-2x-18+2x=6}\implies -18\ne 6\impliedby \stackrel{\textit{system is inconsistent}}{\textit{no solutions}}$

if you set both equations to a y = mx+b, you'll notice the slope is the same, meaning the lines are parallel to each other, thus they never meet.