Follow the hint and solve the system 2x 2y 18 4x 4y 260

Follow the hint and solve the system.

2^x + 2^y = 18

4^x + 4^y = 260 [Hint: Note that 4^x = 2^2x = (2^x)^2.]

Solution

2^x + 2^y=18   => 2^x=18-2^y

4^x + 4^y=260

4^x= 2^2x    and 4^y=2^2y

2^2x + 2^2y=260

(2^x)² + 2^2y=260

(18-2²)² + 2^2y-260=0

324 -36*2^y + 2^2y + 2^2y-260=0

64 -36*2^y +2*2^2y=0

2^2y -18* 2^y +32=0

Let 2^y=t   therefore 2^2y=t²

t² -18t +32=0

(t-2)(t-16)=0

t=2 and t=16

2^y =2   and 2^y =16   =>2^y =2⁴ => y=4

y=1 and y=4

And 2^x+2^y=18

      2^x +2¹=18      and 2^x + 2⁴=18

     2^x=16          and     2^x=2

    x=4 and x=1

therefore the solutions are

(1,4) and (4,1)