Solve for x and y: 4x + 6y = 3xy, 8x + 9y = 5xy

1.	Solve for x and y: 4x + 6y = 3xy, 8x + 9y = 5xy
Answer» The given equations are: 4x + 6y = 3xy …..(i) 8x + 9y = 5xy ……(ii) From equation (i), we have: \(\frac{4x+6y}{xy}=3\) ⇒ 4/y + 6/x = 3 ……(iii) For equation (ii), we have: \(\frac{ax+9y}{xy}=5\) ⇒ 8/y + 9/x = 5 ……(iv) On substituting 1/y = v and 1/x = u, we get: 4v + 6u = 3 ……(v) 8v + 9u = 5 …….(vi) On multiplying (v) by 9 and (vi) by 6, we get: 36v + 54u = 27 ….(vii) 48v + 54u = 30 ….(viii) On subtracting (vii) from (viii), we get: 12v = 3 ⇒ v = 3/12 = 1/4 ⇒ 1/y = 1/4 ⇒ y = 4 On substituting y =4 in (iii), we get: 4/4 + 6/x = 3 ⇒ 1 + 6/x = 3 ⇒ 6/x = (3 – 1) = 2 ⇒ 2x = 6 ⇒ x = 6/2 = 3 Hence, the required solution is x = 3 and y = 4.

Answer»

The given equations are:

4x + 6y = 3xy …..(i)

8x + 9y = 5xy ……(ii)

From equation (i), we have:

\(\frac{4x+6y}{xy}=3\)

⇒ 4/y + 6/x = 3 ……(iii)

For equation (ii), we have:

\(\frac{ax+9y}{xy}=5\)

⇒ 8/y + 9/x = 5 ……(iv)

On substituting 1/y = v and 1/x = u, we get:

4v + 6u = 3 ……(v)

8v + 9u = 5 …….(vi)

On multiplying (v) by 9 and (vi) by 6, we get:

36v + 54u = 27 ….(vii)

48v + 54u = 30 ….(viii)

On subtracting (vii) from (viii), we get:

12v = 3 ⇒ v = 3/12 = 1/4

⇒ 1/y = 1/4 ⇒ y = 4

On substituting y =4 in (iii), we get:

4/4 + 6/x = 3

⇒ 1 + 6/x = 3

⇒ 6/x = (3 – 1) = 2

⇒ 2x = 6 ⇒ x = 6/2 = 3

Hence, the required solution is x = 3 and y = 4.

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