Question

Solve the equations ${\log }_{100}|x+y|=\frac{1}{2},{\log }_{10}y-{\log }_{10}|x|={\log }_{100}4$ for $x$ and $y$.

Answer 1

$|x+y|={100}^{1/2}=10$

${\log }_{10}y-{\log }_{10}\left|x\right|={\log }_{{10}^2}{2}^2=\frac{2}{2}{\log }_{10}2$

$\Rightarrow {\log }_{10}\left(\frac{y}{\left|x\right|}\right)={\log }_{10}2\Rightarrow y=2\left|x\right|$

$\therefore |x+2\left|x\right||=10$

for, $x>0$

$|x+2x|=10\Rightarrow \left|3x\right|=10$

$\Rightarrow 3\left|x\right|=10\Rightarrow 3x=10$

$\Rightarrow x=\frac{10}{3}$ and $y=2\times \frac{10}{3}=\frac{20}{3}$

for, $x<0$

$|x+2(-x\left)\right|=10\Rightarrow |-x|=10$

$\Rightarrow -x=10\Rightarrow x=-10$ and $y=2\left|x\right|=2\times 10=20$

$\therefore x,y\equiv (\frac{10}{3},\frac{20}{3})$ and $(-10,20)$