Multiple choice

Solve the following equations: $\cfrac{7}{{3}^{x}}-\cfrac{6}{{2}^{y}}=15$; $\cfrac{8}{{3}^{x}}=\cfrac{9}{{2}^{y}}$

  1. $(-2 , -3)$
  2. $(-1 , 2)$
  3. $(-4 , 5)$
  4. $(0 , 9)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let u = 1/3^x and v = 1/2^y. Equations: 7u - 6v = 15 and 8u = 9v (so v = 8u/9). Substitute: 7u - 6(8u/9) = 15 => 7u - 16u/3 = 15 => 5u/3 = 15 => u = 9. Since 1/3^x = 9 = 3^2, x = -2. Then 8(9) = 9v, so v = 8. Since 1/2^y = 8 = 2^3, y = -3. Solution is (-2, -3).

AI explanation

Let u = 1/(3^x) and v = 1/(2^y) to transform the equations into 7u - 6v = 15 and 8u = 9v. From the second equation, express u as u = 9v/8 and substitute it into the first equation to get 7(9v/8) - 6v = 15. Simplifying this yields 63v/8 - 48v/8 = 15, so 15v/8 = 15, which means v = 8. Substituting v back gives 8u = 72, so u = 9. Since 1/(3^x) = 9 implies 3^x = 1/9, x = -2; and 1/(2^y) = 8 implies 2^y = 1/8, so y = -3. The solution is (-2, -3).

Keep practicing — related questions