**Q55.**

Given the two equations:

I. $$x^2 = 9^2$$

II. $$(y - 8)^2 = 9$$

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

(a) $$x y$$

(b) $$x \geq y$$

(c) $$y x$$

(d) $$y \geq x$$

Question

**Q55.**

Given the two equations:

I. $$x^2 = 9^2$$

II. $$(y - 8)^2 = 9$$

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

(a) $$x > y$$

(b) $$x \geq y$$

(c) $$y > x$$

(d) $$y \geq x$$

Accepted Answer

**Solution:**

1. From equation I: $$x^2 = 9^2$$
   - Taking the square root on both sides, we get:
     $$x = \pm 9$$
   - Typically, for comparison, we consider the positive root, so $$x = 9$$.

2. From equation II: $$(y - 8)^2 = 9$$
   - Taking the square root on both sides:
     $$y - 8 = \pm 3$$
   - Therefore, $$y = 8 + 3 = 11$$ or $$y = 8 - 3 = 5$$.

3. Comparing the possible values:
   - If $$y = 11$$, then $$y > x$$ since $$11 > 9$$.
   - If $$y = 5$$, then $$x > y$$ since $$9 > 5$$.

4. Since the question does not specify which root to take for *y*, but the standard approach is to consider the positive root for *x* and the positive root for *y - 8* (which is 3), the most reasonable conclusion is:
   $$y = 11 > x = 9$$.

**Answer:** (c) $$y > x$$

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