One day Kim bought 3 packets of rice and 6 cakes for $33. The next day she buys 2 packets of rice and 6 cakes and the bill came to $30 . How much does a packet of rice cost and how much is a cake? 

To solve this problem, let's define the variables:

- Let $ r $ represent the cost of one packet of rice in dollars.
- Let $ c $ represent the cost of one cake in dollars.

We are given two situations that can be translated into algebraic equations:

1. **First situation**: Kim bought 3 packets of rice and 6 cakes for $33.
   \[
   3r + 6c = 33
   \]

2. **Second situation**: Kim bought 2 packets of rice and 6 cakes for $30.
   \[
   2r + 6c = 30
   \]

Now we have a system of two linear equations:

\[
\begin{align*}
3r + 6c &= 33 \quad \text{(Equation 1)} \\
2r + 6c &= 30 \quad \text{(Equation 2)}
\end{align*}
\]

### Step 1: Subtract Equation 2 from Equation 1

By subtracting the second equation from the first, we can eliminate $ c $ and solve for $ r $:

\[
(3r + 6c) - (2r + 6c) = 33 - 30
\]

Simplify the equation:

\[
3r - 2r + 6c - 6c = 3
\]

\[
r = 3
\]

So, the cost of one packet of rice, $ r $, is **$3**.

### Step 2: Substitute the value of $ r $ into one of the equations

Let's substitute $ r = 3 $ into Equation 2 to solve for $ c $:

\[
2r + 6c = 30
\]

Substitute $ r = 3 $:

\[
2(3) + 6c = 30
\]

Simplify:

\[
6 + 6c = 30
\]

Subtract 6 from both sides:

\[
6c = 24
\]

Divide by 6:

\[
c = 4
\]

So, the cost of one cake, $ c $, is **$4**.

### Summary

- **Cost of one packet of rice**: $3
- **Cost of one cake**: $4

Would you like further details on these steps, or do you have any more questions?

Here are some related questions you might find interesting:
1. If Kim bought 5 packets of rice and 10 cakes, how much would it cost?
2. How much would 7 packets of rice and 2 cakes cost?
3. If the price of cakes increased by 20%, how would that affect the total cost in each scenario?
4. If Kim only bought cakes on the third day and spent $24, how many cakes did she buy?
5. What is the difference in cost if Kim buys 8 packets of rice instead of 4?

**Tip:** When dealing with systems of equations, always look for ways to eliminate one variable by adding or subtracting equations. This simplifies finding the value of the remaining variables!

Learn how to solve a problem involving the cost of rice and cakes using systems of linear equations. Understand step-by-step how to determine the cost of one packet of rice and one cake based on two different shopping scenarios. Suitable for students in Grades 7-9.

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