Alg3.3.7

Solve $2^{x+3}=4^{y-2}\,$ and $3^{x-2}=9^{3y-2x}\,$

Simplifying the given equations,

$2^{x+3}=(2^2)^{y-2},3^{x-2}=(3^2)^{3y-2x}\,$

$2^{x+3}=2^{2y-4},3^{x-2}=3^{6y-4x}\,$

$x+3=2y-4,x-2=6y-4x\,$

$x-2y=-7,5x-6y=2\,$

Solving the two equations,

$5(x-2y+7)-(5x-6y-2)=0,5x-10y+35-5x+6y+2=0,-4y+45=0,y=\frac{37}{4}\,$

$x-2\frac{37}{4}=-7,x=\frac{23}{2}\,$

Hence the solution is $x=\frac{23}{2},y=\frac{37}{4}\,$