Given the equations:

I: $$x^2 - 24x + 143 = 0$$

II: $$y^2 - 13y + 22 = 0$$

Which of the following relations between *x* and *y* is correct?

Question

Given the equations:

I: $$x^2 - 24x + 143 = 0$$

II: $$y^2 - 13y + 22 = 0$$

Which of the following relations between *x* and *y* is correct?

Accepted Answer

**Solution:**

1. Consider the first quadratic equation:
   $$x^2 - 24x + 143 = 0$$

2. Factorize the quadratic:
   - Find two numbers whose product is 143 and sum is 24.
   - These numbers are 11 and 13.

3. Rewrite the equation:
   $$x^2 - 11x - 13x + 143 = 0$$

4. Group terms:
   $$x(x - 11) - 13(x - 11) = 0$$

5. Factor out common terms:
   $$(x - 11)(x - 13) = 0$$

6. So, the roots are:
   $$x = 11 	ext{ or } 13$$

7. Now, consider the second quadratic equation:
   $$y^2 - 13y + 22 = 0$$

8. Factorize similarly:
   - Two numbers with product 22 and sum 13 are 11 and 2.

9. Rewrite the equation:
   $$y^2 - 11y - 2y + 22 = 0$$

10. Group terms:
    $$y(y - 11) - 2(y - 11) = 0$$

11. Factor out common terms:
    $$(y - 11)(y - 2) = 0$$

12. Roots are:
    $$y = 11 	ext{ or } 2$$

13. Comparing possible values:
    - *x* can be 11 or 13
    - *y* can be 11 or 2

14. The smallest *x* is 11 and the largest *y* is 11, so *x* is always greater than or equal to *y*.

**Answer:** $x \geq y$

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