Question

Solve the following equations for $x$ and $y$.
$7^x+5^y=74,7^{x+1}-5^{y+1}=218$

Answer 1

$7^x+5^y=74$ ____ (1)

$7^{x+1}-5^{y+1}=218$

$7^x.7-5^y.5=218$ __ (2)

Let$a=7^x;b=5^y$

a +b = 74

a = 74 - b ___ (3)

7a -5b = 218 ___ (4)

Put (a) in (4)

7 (74 - b) -5b = 218

518 - 7b -5b = 218

518-218 = 12b

b = 300/12

b = 25

Put 'b' in (3)

a = 74 - 25 = 49

but $a=7^x$

$7^x=49=7^2$

$\therefore x=2$

b = 25

$5^y=5^2$

$\therefore y=2$