Given the two equations:

(I) $$ x^3 - 119 = 2078$ $

(II) $ $y^2 - 169 = 0 $$

Determine the correct relationship between $ x $ and $ y $:

Question

Given the two equations:

(I) $$ x^3 - 119 = 2078$ $

(II) $ $y^2 - 169 = 0 $$

Determine the correct relationship between $ x $ and $ y $:

Accepted Answer

**Solution:**

1. Solve equation (I):
   $$
  $ x^3 - 119 = 2078$
   $$
   Add 119 to both sides:
   $$
 $  x^3 = 2078 + 119 = 2197$
   $$
   Now, find the cube root of 2197:
   $$
 $  x = \sqrt[3]{2197} = 13$
   $$

2. Solve equation (II):
   $$
 $  y^2 - 169 = 0$
   $$
   Add 169 to both sides:
   $$
 $  y^2 = 169 $
   $$
   Taking square root:
   $$
   y = \pm 13
   $$

3. Compare $ x $ and $ y $:
   - Since $ x = 13 $
   - $ y $ can be either $ 13 $ or $ -13 $

4. Considering both values of $ y $:
   - If $ y = 13 $, then $ x = y $
   - If $ y = -13 $, then $ x > y $

Therefore, the relationship $ $x \geq y $$ always holds true.

**Answer:** (b) $$ x \geq y $$

Answer

Answer

Answer

Answer