Exponential Recap II

$\large 2^{x+y}\cdot 4^x=64$ $\large 16^{x+y}\cdot 2^{-y}=16$

1.) Make the bases equal.

$2^{x+y}\cdot 2^{2x}=2^6$

$2^{4x+4y}\cdot 2^{-y}=2^4$

2.) Once all the bases are equal, take the exponents as linear equations, while still applying the rules of exponents.

$3x+y=6$

$4x+3y=4$

3.) Solve like how you would normally solve linear equations.. In my case, I used the substitution method.

$3x+y=6$ >>> $y=-3x+6$

$4x+3(-3x+6)=4$

$-5x+18=4$

$5x=-14$

$x =$ $\dfrac{14}{5}$

$y=-3\left(\frac{14}{5}\right)+6$

$y=-\frac{12}{5}$

4.) Check if both values are correct by substituting the variables as their respective values.

(The complete procedure isn't shown but both values are correct.)

$64=64$

$16=16$

5.) Once you've solved for the values and checked if they are the correct ones, add the values of both variables, then add the numerator and denominator of the sum, as instructed.

$\frac{14}{5}-\frac{12}{5}=\frac{2}{5}$

$2+5 = 7$

∴ The answer is $7$ .