Question

Solve the following systems of equations.
$8^{lo{g}_2(x-4y)}=1,4^{x-2y}-{7.2}^{x-2y}=8$

Answer 1

Solve $8^{lo{g}_2(x-4y)}=1...\left(1\right)$

$4^{x-2y}-{7.2}^{x-2y}=8...\left(2\right)$

by eq(1) $8^{lo{g}_2(x-4y)}=1$

$2^{3lo{g}_2(x-4y)}=1$

$(x-4y{)}^3=1$

$\Rightarrow (x-4y)=1...\left(3\right)$

by eq.(2)

$4^{x-2y}-{7.2}^{x-2y}=8$

$(2^{(x-2y)}{)}^2-{7.2}^{(x-2y)}-8=0$

Let $2^{(x-2y)}=t$

$\Rightarrow t^2-7t-8=0$

$(t+1)(t-8)=0$

$t=1,t=8$

Let $t=8=2^{(x-2y)}$

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

$\Rightarrow x-2y=3...\left(4\right)$

We can't take $t=-1$

$\therefore t=2^{(x-2y)}$ is never be negative

by eq. (3)-(4)

$-2y=-2\Rightarrow y=1$

and by eq.(4) $x=5$ sol