Question

If 2^a + 3^b = 17 and 2^a + 2 – 3^{b + 1} = 5, then find the value of a and b.

Answer 1

2^a + 3^b = 17......... …(1)

2^{a + 2} – 3^{b + 1} = 5

\(\Rightarrow\) 2².2^a - 3.3^b = 5

\(\Rightarrow\) 4.2^a – 3.3^b = 5.............(ii)

Let 2^a = x and 3^b = y. Then the system of equations is 

x + y = 17 .......…(3) 

4x – 3y = 5............(4)

Multiplying equations (3) by 3 and adding to equation (4), we get

x = \(\frac{56}{7}\) = 8

Putting x = 8 in (3), we get

8 + y = 17 \(\Rightarrow\) y = 17 - 8 = 9

\(\Rightarrow\) 2^a = 8 and 3^b = 9 \(\Rightarrow\) 2^a =2³ and 3^b = 3²

\(\Rightarrow\) a= 3 and b = 2