Elimu school bought 25 textbooks and 35 exercise books for sh. 13500. Soma school bought 21 textbooks and 38 exercise books for sh. 12200. Find the cost of one textbook and one exercise book.

For Elimu school:
25 textbooks and 35 exercise books cost sh. 13500.

$$25x + 35y = 13500 \quad \text{(Equation 1)}$$

For Soma school:
Soma school bought 21 textbooks and 38 exercise books.
They spent sh. 1300 less than Elimu school.
Elimu school spent sh. 13500, so Soma school spent $13500 - 1300 = 12200$ sh.

$$21x + 38y = 12200 \quad \text{(Equation 2)}$$

Multiply Equation 1 by 21 and Equation 2 by 25 to eliminate $x$.
Equation 1 $\times$ 21:

$$21(25x) + 21(35y) = 21(13500)$$ $$525x + 735y = 283500 \quad \text{(Equation 3)}$$

Equation 2 $\times$ 25:

$$25(21x) + 25(38y) = 25(12200)$$ $$525x + 950y = 305000 \quad \text{(Equation 4)}$$

$$(525x + 950y) - (525x + 735y) = 305000 - 283500$$ $$215y = 21500$$ $$y = \frac{21500}{215}$$ $$y = 100$$

The cost of one exercise book is sh. 100.

$$25x + 35(100) = 13500$$ $$25x + 3500 = 13500$$ $$25x = 13500 - 3500$$ $$25x = 10000$$ $$x = \frac{10000}{25}$$ $$x = 400$$

The cost of one textbook is sh. 400.

The cost of one textbook is $\boxed{\text{sh. } 400}$.
The cost of one exercise book is $\boxed{\text{sh. } 100}$.