\$\$eginalign*2left( x-frac13 ight) -frac32left( y-frac16 ight)&=0 ag 1\ 3left( y-frac12 ight) +frac83left( x-frac16 ight)&=0 ag 2\ endalign*\$\$Consider the system of equations above. If \$(x,y)\$ is the systems to the system, climate what is the worth of the amount of \$x\$ and also \$y\$?

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 \ endpmatrixeginpmatrixx \ 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. \$\$