Question

Solve the following simultaneous equations.

${\displaystyle \frac{2}{x+y}-\frac{3}{x-y}}$ = 15; ${\displaystyle \frac{8}{x+y}+\frac{5}{x-y}}$ = 77

Answer 1

${\displaystyle \frac{2}{x+y}-\frac{3}{x-y}}$ = 15  ....(i)

${\displaystyle \frac{8}{x+y}+\frac{5}{x-y}}$ = 77 ....(ii)

Replacing ${\displaystyle \left(\frac{1}{x+y}\right)}$ by a and ${\displaystyle \frac{1}{x-y}}$ by b in equation (i) and equation (ii), we get 

2a - 3b = 15   ....(iii)

8a + 5b = 77  ....(iv)

Multiplying equation (iii) by 5 and equation (iv) by 3

10a – 15b = 75  ....(v)

24a + 15b = 231  ...(vi)

Now adding equations (v) and (vi), we get

⇒ 10a + 24a = 75 + 231

⇒ 34a = 306

∴ a = ${\displaystyle \frac{306}{34}}$

∴ a = 9

Substituting, a = 9 in equation (iii), we get

⇒ 2(9) – 3b = 15

⇒ 3b = 18 – 15

∴ 3b = 3

∴ b = 1

By reversing the values of a and b, we get

${\displaystyle \frac{1}{x+y}}$ = 9 and ${\displaystyle \frac{1}{x-y}}$ = 1

∴ x + y = 9  ....(vii)

And x – y = 1  ....(viii)

Adding equations (vii) and (viii), we get

∴ 2x = 10

x = 5

Substituting, x = 5 in equations (vii), we get

5 + y = 9

∴ y = 4

As a result, the above equation's solution is x = 5, y = 4.