Question

x-y=36 , xy=1440 find x and y

Answer 1

Required Answer:-

Given:

$\dag \: \begin{cases}\sf x - y = 36 \\ \sf xy = 1440 \end{cases}$

To Find:

• The values of x and y.

Solution:

This problem can be solved in two ways.

1. First approach.

Given that,

$\sf \implies x - y = 36 \: ...(i)$

Squaring both sides, we get,

$\sf \implies (x - y)^{2} = {36}^{2}$

Using identity (a - b)² = a² - 2ab + b², we get,

$\sf \implies {x}^{2} - 2xy + {y}^{2} =1296$

Adding 4xy on both sides, we get,

$\sf \implies {x}^{2} + 2xy + {y}^{2} =1296 + 4xy$

Using identity (a + b)² = a² + 2ab + b², we get,

$\sf \implies {(x + y)}^{2} = 1296 + 4xy$

Substituting the value of xy, we get,

$\sf \implies {(x + y)}^{2} = 1296 + 4 \times 1440$

$\sf \implies {(x + y)}^{2} = 1296 + 5760$

$\sf \implies {(x + y)}^{2} =7056$

$\sf \implies (x + y)= \sqrt{7056}$

$\sf \implies (x + y)= \pm 84$

Therefore,

$\sf \implies (x + y)=84 \: ...(ii)$

$\sf \implies (x + y)= -84 \: ...(iii)$

Adding equations (i) and (ii), we get,

$\sf \implies 2x = 120$

$\sf \implies x = 60$

Therefore,

$\sf \implies x - y = 36$

$\sf \implies 60 - y = 36$

$\sf \implies y = 60 - 36$

$\sf \implies y =24$

From here,

$\sf \implies x = 60 \: and \: y = 24$

Again, Adding equations (i) and (iii), we get,

$\sf \implies 2x = - 48$

$\sf \implies x = -24$

Therefore,

$\sf \implies x - y = 36$

$\sf \implies - 24- y = 36$

$\sf \implies - y = 36 + 24$

$\sf \implies - y = 60$

$\sf \implies y = -60$

Thus,

$\sf\implies x = 60, - 24 \ and \ y = - 60,24$

2. Second Approach.

Given that,

$\sf \implies x - y = 36$

$\sf \implies x = y + 36$

Also,

$\sf \implies xy = 1440$

Substituting the values of x, we get,

$\sf \implies y(y + 36) = 1440$

$\sf \implies {y}^{2} + 36y= 1440$

$\sf \implies {y}^{2} + 36y - 1440 = 0$

$\sf \implies {y}^{2} - 24y + 60y- 1440 = 0$

$\sf \implies y(y- 24)+ 60(y- 24)= 0$

$\sf \implies (y+ 60)(y- 24)= 0$

By zero product rule,

$\sf \implies (y+ 60) = 0 \: or \: (y- 24)= 0$

Therefore,

$\sf \implies y = -60 ,24$

Therefore, when y = -60,

$\sf \implies x = y + 36$

$\sf \implies x = - 60+ 36$

$\sf \implies x = -24$

Also, when y = 24,

$\sf \implies x = y + 36$

$\sf \implies x = 24+ 36$

$\sf \implies x =60$

Therefore,

$\sf \implies x= 60 , - 24$

So,

$\sf \implies x = 60, - 24 \ and \ y = - 60,24$

Answer:

$\dag \: \begin{cases}\sf x = 60, - 24 \\ \sf y = - 60,24\end{cases}$