83. Tenglamalar sistemasini yeching: 7/(x+y) − 6/(x−y) = −1; 1/(x+y) + 1/(x−y) = 10/21.

Belgilaymiz: \(a=\frac{1}{x+y}\), \(b=\frac{1}{x-y}\). U holda: \[\begin{cases}7a-6b=-1,\\a+b=\frac{10}{21}.\end{cases}\] Ikkinchi tenglamadan \(a=\frac{10}{21}-b\). Birinchi tenglamaga qo‘yamiz: \[7\left(\frac{10}{21}-b\right)-6b=-1.\] Bundan \(\frac{10}{3}-13b=-1\), ya’ni \(b=\frac{4}{13}\). Shunda \(a=\frac{10}{21}-\frac{4}{13}=\frac{46}{273}\). Demak, \[x+y=\frac{273}{46},\qquad x-y=\frac{13}{4}.\] Qo‘shib va ayirib: \[2x=\frac{273}{46}+\frac{13}{4}=\frac{839}{92},\qquad x=\frac{839}{184};\] \[2y=\frac{273}{46}-\frac{13}{4}=\frac{313}{92},\qquad y=\frac{313}{184}.\]

Javob: \(x=\frac{839}{184},\ y=\frac{313}{184}\).