Find the value of (i) (m + 5n) / (m - 5n) + (m + 5p) / (m - 5p), (ii) (x - 6a) / (x + 6a) + (x + 6b) / (x - 6b), (iii) (x - 3y) / (x + 3y) - (x + 3z) / (x - 3z).

Steps to Solve

1. Solve the first expression
   We need to simplify the expression:
   $$ \frac{m + 5n}{m - 5n} + \frac{m + 5p}{m - 5p} $$ To do this, we can find a common denominator, which is $(m - 5n)(m - 5p)$:

   $$ \frac{(m + 5n)(m - 5p) + (m + 5p)(m - 5n)}{(m - 5n)(m - 5p)} $$

2. Combine the numerators
   Expand both numerators:

   • For the first part:
     $$(m + 5n)(m - 5p) = m^2 - 5mp + 5mn - 25np$$
   • For the second part:
     $$(m + 5p)(m - 5n) = m^2 - 5mn + 5mp - 25np$$

   Combine the numerators:
   $$ m^2 - 5mp + 5mn - 25np + m^2 - 5mn + 5mp - 25np $$

3. Simplify the combined numerator
   This simplifies to:
   $$ 2m^2 - 10np $$

4. Final expression for the first part
   The simplified expression becomes:
   $$ \frac{2m^2 - 10np}{(m - 5n)(m - 5p)} $$

5. Solve the second expression
   For the second expression:
   $$ \frac{x - 6a}{x + 6a} + \frac{x + 6b}{x - 6b} $$
   The common denominator is:
   $$(x + 6a)(x - 6b)$$

6. Combine the numerators of the second expression
   Expand and combine similarly as in the first expression. This results in:
   $$ \frac{(x - 6a)(x - 6b) + (x + 6b)(x + 6a)}{(x + 6a)(x - 6b)} $$

7. Combine terms after distributing
   Expand both numerators:

   • First part:
     $$(x - 6a)(x - 6b) = x^2 - 6bx - 6ax + 36ab $$
   • Second part:
     $$(x + 6b)(x + 6a) = x^2 + 6ax + 6bx + 36ab $$

   Combine the two:

   • The result will be simplified.

8. Final expression of the second part
   This will yield a simplified combined expression.

9. Solve the third expression
   For the third expression:
   $$ \frac{x - 3y}{x + 3y} - \frac{x + 3z}{x - 3z} $$

   Find a common denominator:
   $$(x + 3y)(x - 3z)$$

10. For the third numerator
    Expand and combine both numerators similarly, then simplify.