Question

Solve the following systems of equations.
$4^{(x-y)2-1}=1,5^{x+y}=125$

Answer 1

$4^{(x-y)2-1}=\mathrm{1...}\left(1\right)$

$5^{x+y}=125...\left(2\right)$

in eq (1) $4^{(x-y)2-1}=1$

$\frac{4^{(x-y)2}}{4}=1\Rightarrow 4^{(x-y)2}=4$

$4^{(x-y)2}=4$

taking log both sides

$2(x-y)log4=log4$

$2(x-y)=1$

$x-y=\frac{1}{2}...\left(3\right)$

in eq (2)

$5^{x+y}=125$

$5^{x+y}=5^3$

taking log both sides

$(x+y)lo{g}_5=3lo{g}_5$

$x+y=3...\left(4\right)$

by eq (3) & (4)

$x-y=\frac{1}{2}$

$x+y=3$

sun of eq (3) & (4)

$2x=3+\frac{1}{2}$

$\Rightarrow x=\frac{7}{4}$

putting value of x

in eq (3)

$y=\frac{7}{4}-\frac{1}{2}$

$y=5/4$