Question

2^x + 3^y = 17 and 2^x+2 - 3^y+1 = 5

Answer 1

The given pair of linear equations is;

2^x + 3^y = 17 ----------(1)

and

2^(x+2) - 3^(y+1) = 5

(2^x)(2^2) - (3^y)(3^1) = 5

4(2^x) - 3(3^x) = 5 ------------(2)

Let, 2^x = a and 3^y = b

Thus, eq-(1) would become;

a + b = 17

OR

a = 17 - b ---------(3)

And , eq-(2) would become;

4a - 3b = 5 ---------(4)

Now,

Putting the value of a=17-b in eq-(4) ,

We get;

=> 4a - 3b = 5

=> 4(17 - b) - 3b = 5

=> 68 - 4b - 3b = 5

=> 4b + 3b = 68 - 5

=> 7b = 63

=> b = 63/7

=> b = 9

Also, b = 3^y

Thus, we have;

=> b = 9

=> 3^y = 9

=> 3^y = 3^2

=> y = 2

Now,

Putting the value b=9 in eq-(3),

We get ;

=> a = 17 - b

=> a = 17 - 9

=> a = 8

Also, a = 2^x

Thus, we have;

=> a = 8

=> 2^x = 8

=> 2^x = 2^3

=> x = 3

Hence,

The solution of the given pair of linear equations is , x = 3 and y = 2

ie; (3,2)