Question

Solve the system of equations $\{\begin{array}{c}\frac{1}{x}+\frac{1}{y}=\frac{3}{2}\\ \frac{1}{x^2}+\frac{1}{y^2}=\frac{5}{4}\end{array}$

• A. $(1,2)$
• B. $(1,3)$
• C. $(2,3)$
• D. $(2,2)$

Answer 1

The correct option is A $(1,2)$

$\frac{1}{x}+\frac{1}{y}=\frac{3}{2},\frac{1}{x^2}+\frac{1}{y^2}=\frac{5}{4},\frac{1}{x}=a,\frac{1}{y}=b,a+b=\frac{3}{2},a^2+b^2=\frac{5}{4},(a+b{)}^2={\left(\frac{3}{2}\right)}^2=a^2+b^2+2ab=\frac{9}{4},\frac{5}{4}+2ab=\frac{9}{4},2ab=1,b=\frac{1}{2a},a+\frac{1}{2a}=\frac{3}{2},2a^2+1=3a,2a^2-2a-a+1=0,(2a-1\left)\right(a-1)=0,a=\frac{1}{2}or1,b=\frac{1}{2}or1,xandy=1and2$