If 3x + y = 81 and 81x – y = 3, then the value of x and y are(a) \(\

1.	If 3x + y = 81 and 81x – y = 3, then the value of x and y are(a) \(\frac{17}{8},\frac98\)(b) \(\frac{17}{8},\frac{15}{8}\)(c) \(\frac{17}{8},\frac{11}8\)(d) \(\frac{15}{8},\frac{11}8\)
Answer» (b) \(\frac{17}{8}\), \(\frac{15}{8}\) 3^{x + y} = 81 ⇒ 3^{x + y} = 3⁴ = x + y = 4 ... (i) 81^{x – y}= 3 ⇒ (3⁴)^{x – y} = 3¹ ⇒ 4x – 4y = 1 ... (ii) Eqn (i) × 4 + Eqn (ii) gives 4x + 4y + 4x – 4y = 16 + 1 ⇒ 8x = 17 ⇒ x = \(\frac{17}{8}\) Putting x = \(\frac{17}{8}\) in (i), we get \(\frac{17}{8}\) + y = 4 ⇒ y = 4 - \(\frac{17}{8}\) = \(\frac{15}{8}\) ∴ x = \(\frac{17}{8}\), y = \(\frac{15}{8}\).

If 3x + y = 81 and 81x – y = 3, then the value of x and y are(a) \(\frac{17}{8},\frac98\)(b) \(\frac{17}{8},\frac{15}{8}\)(c) \(\frac{17}{8},\frac{11}8\)(d) \(\frac{15}{8},\frac{11}8\)

Answer»

(b) \(\frac{17}{8}\), \(\frac{15}{8}\)

3^{x + y} = 81 ⇒ 3^{x + y} = 3⁴ = x + y = 4 ... (i)

81^{x – y}= 3 ⇒ (3⁴)^{x – y} = 3¹

⇒ 4x – 4y = 1 ... (ii)

Eqn (i) × 4 + Eqn (ii) gives

4x + 4y + 4x – 4y = 16 + 1

⇒ 8x = 17 ⇒ x = \(\frac{17}{8}\)

Putting x = \(\frac{17}{8}\) in (i), we get \(\frac{17}{8}\) + y = 4

⇒ y = 4 - \(\frac{17}{8}\) = \(\frac{15}{8}\)

∴ x = \(\frac{17}{8}\), y = \(\frac{15}{8}\).