निम्नलिखित समीरकरण निकाय को x और y के लिये हल कीजिए :
`a(x+y)+b(x-y) =a^(2) - ab+b^(2) ` और `a(x+y) -b(x-y)=a^(2) +ab +b^(2)`

Explanation

दी समीकरण :
`ax + ay + bx - by = a^(2) - ab +b^(2)`
`rArr (a+b)x+(a-b)y =a^(2) - ab +b^(2)`
और `ax + ay - bx + by =a^(2) +ab + b^(2)`
`rArr" "(a-b)x + (a+b)y = a^(2)+ab+b^(2)`
समीकरण (1 ) और (2 ) में वक्र गुणन विधि से

`(x)/((a-b)xx[-(a^(2)+ab+b^(2))] -(a+b) xx[-(a^(2) - ab+b^(2))]) = (y)/([-(a^(2) -ab+b^(2))]xx(a-b)-[-(a^(2) +ab+b^(2))xx(a+b)]) = (1)/((axxb)(a+b) -(a-b)(a-b))`
`rArr (x)/(-(a^(3)- b^(3))+(a^(3) +b^(3))) = (y)/(-(a^(3)-a^(2)b+ab^(2) -a^(2)b+ab^(2) -b^(3))+(a^(3)+a^(2)b+ab^(2)+a^(2)b+ab^(2)+b^(3))`
` = (1)/((a+b)^(2)-(a-b)^(2))`
`rArr (x)/(2b^(3)) = (y)/(2a^(2)b - 2ab^(2)+b^(3)+2a^(2)b+2ab^(2) +b^(3)) = (1)/(4ab)`
`rArr (x)/(2b^(3)) = (y)/(4a^(2)b +2b^(3)) = (1)/(4ab)`
`rArr (x)/(2b^(3)) = (y)/(2b(2a^(2)+b^(2)) =(1)/(4ab)`
जब `(x)/(2b^(3)) = (1)/(4ab) " "rArr " "x = (2b^(3))/(4ab) " "rArr" "x =(b^(2))/(2a)`
और mr`(y)/(2b(2a^(2) +b^(2))) = (1)/(4ab) rArr y = (2b(2a^(2) +b^(2)))/(4ab) rArr y = (2a^(2) +b^(2))/(2a)`
अतः हल `{:(x = (b^(2))/(2a)),(y = (2a^(2) +b^(2))/(2a)):}}`है |