Calculate the value of x and y if 27x ÷ 81x+2y = 9, x + 4y = 0

Calculate the value of x and y if 27x ÷ 81x+2y = 9, x + 4y = 0

Answer Details

${27}^x÷{81}^{(x+2y)}=9$
$\left(27\right)x=9\times {81}^{(x+2y)}$
$(3^3{)}^x=3^2\times 3^{4(x+2y)}$
$=3^{(2+4x+8y)}$
$3^{3x}=3^{(2+4x+8y)}$
$3x=2+4x+8y$
$3x-4x-8y=2........\left(1\right)$
$x+4y=0........\left(2\right)$
$-4y=2$
$y=(-2)÷4=-\frac{1}{2}$
$y=-\frac{1}{2}$
Substitute the value of y into equation (2)
i.e $x+4y=0$
$x+4(-\frac{1}{2})=0$
$x-2=0$
$x=2$
$\therefore x=2,y=-\frac{1}{2}$
Method II
${27}^x÷{31}^{(x+2y)}=9$
$3^{3x}\times 3^{(-4x-8y)}=3^2$
$3^{(3x-8y)}=3^2$
$-x-8y=2........\left(1\right)$
$x+4y=0........\left(2\right)$
$-4y=2$
$y=-\frac{2}{4}=-\frac{1}{2}$
$y=-\frac{1}{2}$
Substitute the value of y into equation 2
$x+4y=0$
$x+4(-1)÷2)=0$
$x-2=0$
$x=2$
$x=2,y=-\frac{1}{2}$