Question

If 2^a + 3^b = 17 and 2^(a+2) - 3^(b+1) =5, then find the value of a & b.

Answer 1

Answer:

a = 3 , b = 2

Step-by-step explanation:

Given :

$2^a + 3^b = 17$  ...(i)

&

$2^{a+2} - 3^{b+1} = 5$  ...(ii)

To Find :

The values of a & b,.

Solution :

Let $2^a = x$ & $3^b = y$

Then,

(i) ⇒$2^a + 3^b = 17$

⇒ x + y = 17  ...(iii)

(ii) ⇒ $2^{a + 2} - 3^{b + 1} = 5$

⇒ $(2^2)(2^{a}) - (3^1)(3^{b}) = 5$

⇒ $4(2^{a})- 3(3^{b}) = 5$

⇒ 4x - 3y = 5 ...(iv)

Then,

By adding both the equation obtained from (i) & (ii)

i.e., 3 × (i) + (ii)

⇒ 3(x + y) + (4x - 3y) = 3(17) + 5

⇒ 3x + 3y + 4x - 3y = 51 + 5

⇒ 7x = 56

⇒ x = 8,.

⇒ $2^a = 8$

⇒ $2^a = 2^3$

⇒ a = 3,.

_

By substituting value of x in (iii),

We get,

⇒ x + y = 17

⇒ 8 + y = 17

⇒ y = 17 - 8

⇒ y = 9,

⇒ $3^b = 9$

⇒ $3^b = 3^2$

⇒ $b = 2$

∴ a = 3 , b = 2

Answer 2

Answer:

a = 3 , b = 2

Step-by-step explanation:

Given :

2^a + 3^b = 17 ...(i)

&

2^{a+2} - 3^{b+1} = 5 ...(ii)

To Find :

The values of a & b,.

Solution :

Let 2^a = x

&

3^b = y

Then,

(i) ⇒2^a + 3^b = 17

⇒ x + y = 17 ...(iii)

(ii) ⇒ 2^{a + 2} - 3^{b + 1} = 5

⇒ (2^2)(2^{a}) - (3^1)(3^{b}) = 5

⇒ 4(2^{a})- 3(3^{b}) = 5

⇒ 4x - 3y = 5 ...(iv)

Then,

By adding both the equation obtained from (i) & (ii)

i.e., 3 × (i) + (ii)

⇒ 3(x + y) + (4x - 3y) = 3(17) + 5

⇒ 3x + 3y + 4x - 3y = 51 + 5

⇒ 7x = 56

⇒ x = 8,.

⇒ 2^a = 8

⇒ 2^a = 2^3

•°• a = 3,.

_

By substituting value of x in (iii),

We get,

⇒ x + y = 17

⇒ 8 + y = 17

⇒ y = 17 - 8

⇒ y = 9,

⇒ 3^b = 9

⇒ 3^b = 3^2

•°• b = 2

$\red{.°.a = 3 , b = 2}$