For the real numbers \(p\) \(=\) 22 and \(q\) \(=\) 58, if \(p\) is increased by \(u\) \(=\) \(2\) and \(q\) is decreased by \(v\) \(=\) \(7\), then show that the increment in product is always equal to \(uq\) \(-\) \(pv\) \(-\) \(uv\).
Proof:
Given, \(p\) \(=\) 22, \(q\) \(=\) 58, \(u\) \(=\) \(2\), and \(v\) \(=\) \(7\).
Their product \(pq\) \(=\) .
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.