For the real numbers \(m\) \(=\) 25 and \(n\) \(=\) 58, if \(m\) is increased by \(u\) \(=\) \(4\) and \(n\) is decreased by \(v\) \(=\) \(7\), then show that the increment in product is always equal to \(un\) \(-\) \(mv\) \(-\) \(uv\).
Proof:
Given, \(m\) \(=\) 25, \(n\) \(=\) 58, \(u\) \(=\) \(4\), and \(v\) \(=\) \(7\).
Their product \(mn\) \(=\) .
By the given condition, the new product is given by .
The value of the new product \(=\)
The difference in products \(=\)
By the distributive property, the increment in product is given by .
The increment in product using the obtained property \(=\) .
Difference in product Increment in product
Hence, proved.