Solve the following mathematical problems:

Solve: $x^{2} - x y = 7, y^{2} + x y = 30$ (Equation (সমীকরণ))

Created: 1 year ago | Updated: 1 year ago

Updated: 1 year ago

$G i v e n, x^{2} - y x = 7 . . . . . . . . . . . . . . (i) \Rightarrow x y = x^{2} - 7 \Rightarrow y = \frac{x^{2} - 7}{x} \Rightarrow y = x - \frac{7}{x} . . . . . . . . . (i i)$

Another equation is $y^{2} + x y = 30 . . . . . . . . . . (i i i)$

Adding equation (i) and (iii) we get

$\frac{x^{2} - x y = 7 y^{2} + x y = 30}{x^{2} + y^{2} = 37 . . . . . (i v)}$

Now, putting the value of y in equation (iv) we get

$x^{2} + {(x - \frac{7}{x})}^{2} = 37 \Rightarrow x^{2} + x^{2} - 2 \times x \times \frac{7}{x} + {(\frac{7}{x})}^{2} = 37 \Rightarrow 2 x^{2} - 14 + \frac{49}{x^{2}} = 37 \Rightarrow 2 x^{2} + \frac{49}{x^{2}} = 51 \Rightarrow \frac{2 x^{4} + 49}{x^{2}} = 51 \Rightarrow 2 x^{4} - 51 x^{2} + 49 = 0 \Rightarrow 2 x^{4} - 2 x^{2} - 49 x^{2} + 49 = 0 \Rightarrow 2 x^{2} (x^{2} - 1) - 49 (x^{2} - 1) = 0 \Rightarrow (x^{2} - 1) (2 x^{2} - 49) = 0 H e r e, x^{2} - 1 = 0 \Rightarrow x^{2} = 1 \Rightarrow x = \sqrt{1} ∴ x = 1 A n d, 2 x^{2} - 49 = 0 \Rightarrow x^{2} = \frac{49}{2} ∴ x = \sqrt{\frac{49}{2}} = \frac{7}{\sqrt{2}} T h u s w e g e t z = 1, \frac{7}{\sqrt{2}} W h e n x = 1, t h e n y = 1 - \frac{7}{1} = 1 - 7 = - 6 A n d w h e n x = \frac{7}{\sqrt{2}}, t h e n y = \frac{7}{\sqrt{2}} - \frac{7}{7} = \frac{7}{\sqrt{2}} - \frac{7}{1} \times \frac{\sqrt{2}}{7} = \frac{7}{\sqrt{2}} - \sqrt{2} = \frac{7 - {(\sqrt{2})}^{2}}{\sqrt{2}} = \frac{7 - 2}{\sqrt{2}} = \frac{5}{\sqrt{2}} ∴ (x, y) = (1, - 6) o r (\frac{7}{\sqrt{2}}, \frac{5}{\sqrt{2}}) a n s .$