I"ve provided the solving for $x$, and then, detect $y$ method, however I feel, that there should be a faster way to uncover $x+y$. Go anyone know any type of tricks?

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In general, we should multiply $(1)$ through $a$ and $(2)$ through $b$ for this reason that as soon as the two equations are added we acquire the coefficients the $x$ and also $y$ the same:$$2a+frac83b=3b-frac32a Rightarrow 21a=2b Rightarrow a=2, b=21.$$

Hence:$$60(x+y)=(frac23-frac14)cdot2+(frac32+frac49)cdot21 Rightarrow x+y=frac2536.$$

$$eginalign*2left( x-frac13 ight) -frac32left( y-frac16 ight)&=0 \ 3left( y-frac12 ight) +frac83left( x-frac16 ight)&=0 \ endalign*$$Multiply the first by $2$ and the second by $21$$$eginalign*4 x-3 y&=frac56 \ 56 x+63 y&= frac2456\ endalign*$$Then add the two equations$$60x+60y= frac2506 ightarrow x+y=frac2536$$

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Begin with the mechanism of 2 equations: $$eginalign*2left( x-frac13 ight) -frac32left( y-frac16 ight)&=0, \ 3left( y-frac12 ight) +frac83left( x-frac16 ight)&=0. \ endalign*$$Multiply through the very first equation through $colorblue12$ and also multiply v the second equation through $colorgreen18$ so the we have the right to clear the denominator for both equations: $$eginalign*colorblue4cdot 2left( colorblue3left( x-frac13 ight) ight) - colorblue2cdot frac32left(colorblue6left( y-frac16 ight) ight)&=0, \ colorgreen9cdot 3left( colorgreen2left( y-frac12 ight) ight) +colorgreen3cdot frac83left( colorgreen6left(x-frac16 ight) ight)&=0. \ endalign*$$Now, multiply out the numbers in blue and also in green: $$eginalign*8left( 3 x-1 ight) - 3 left(6y-1 ight)&=0, \ 27 left( 2y-1 ight) + 8 left( 6x-1 ight)&=0. \ endalign*$$Expand: $$eginalign*24x -8 -18y+3 &= 0, \ 54 y - 27 + 48x-8 &= 0. \ endalign*$$Simplify additional as: $$eginalign*24x - 18y &= 8-3=5, \ 48x + 54 y &= 27+8=35. \ endalign*$$Write the 2 equations in procession form: $$eginpmatrix24 & -18 \ 48 & 54 \ endpmatrixeginpmatrixx \ y \ endpmatrix = eginpmatrix5 \ 35 \ endpmatrix. $$Write this as an augmented matrix: $$eginpmatrix24 & -18 & 5\ 48 & 54 & 35 \ endpmatrix $$ and also then row-reduce (multiply the first row through $-2$ and also then include the equivalent numbers come the 2nd row): $$eginpmatrix24 & -18 & 5\ 0 & 90 & 25 \ endpmatrix. $$ divide the second row through $5$ to get: $$eginpmatrix24 & -18 & 5\ 0 & 18 & 5 \ endpmatrix. $$Add the 2nd row to the first row: $$eginpmatrix24 & 0 & 10\ 0 & 18 & 5 \ endpmatrix. $$Divide the first row through $24$ and also divide the second row by $18$: $$eginpmatrix1 & 0 & frac512\ 0 & 1 & frac518 \ endpmatrix. $$So $$x= frac512 mbox and y= frac518, $$and we conclude $$oxedx+y = frac512 + frac518 = frac2536. $$