Two kilograms of brand A rice and three kilograms of brand B rice cost Sh 900. One kilogram of brand A rice and two kilograms of brand B rice cost Sh 550. Find the cost of one kilogram of brand A and

Let $A$ be the cost of $1\text{kg}$ of brand A rice.
Let $B$ be the cost of $1\text{kg}$ of brand B rice.

From the first statement: $2\text{kg}$ of brand A and $3\text{kg}$ of brand B cost Sh 900.

$$2A + 3B = 900 \quad (1)$$

From the second statement: $1\text{kg}$ of brand A and $2\text{kg}$ of brand B cost Sh 550.

$$A + 2B = 550 \quad (2)$$

Multiply equation (2) by 2 to make the coefficients of $A$ equal:

$$2(A + 2B) = 2(550)$$ $$2A + 4B = 1100 \quad (3)$$

$$(2A + 4B) - (2A + 3B) = 1100 - 900$$ $$2A + 4B - 2A - 3B = 200$$ $$B = 200$$

The cost of $1\text{kg}$ of brand B rice is Sh 200.

$$A + 2B = 550$$ $$A + 2(200) = 550$$ $$A + 400 = 550$$ $$A = 550 - 400$$ $$A = 150$$

The cost of $1\text{kg}$ of brand A rice is Sh 150.

The cost of $1\text{kg}$ of brand A rice is $\boxed{\text{Sh 150}}$ and the cost of $1\text{kg}$ of brand B rice is $\boxed{\text{Sh 200}}$.