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The problem asks to determine when two stores charge the same rate for dry cleaning. Let's set up the information:

- Store A's price is represented by the expression: \( 15x - 2 \).
- Store B's price is represented by the expression: \( 3(5x + 7) \).

We are tasked with finding when the two stores charge the same rate, which means we need to set the two expressions equal to each other:

\[
15x - 2 = 3(5x + 7)
\]

Now, let's solve the equation step by step.

### Step 1: Expand the expression for Store B.
\[
3(5x + 7) = 15x + 21
\]

So the equation becomes:
\[
15x - 2 = 15x + 21
\]

### Step 2: Subtract \( 15x \) from both sides to eliminate the \( x \) terms.
\[
-2 = 21
\]

### Step 3: Analyze the result.
Since \( -2 \neq 21 \), this equation leads to a contradiction. This means that there is **no solution**, and therefore the two stores **never charge the same rate**.

Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:
1. How do we handle contradictions in equations?
2. What if the prices had different slopes instead of both being \( 15x \)?
3. Can two stores charge the same rate if they have different pricing structures?
4. How do systems of equations work when no solution exists?
5. What types of problems lead to contradictions like this one?

**Tip:** In equations involving two expressions, contradictions usually indicate that the lines or relationships are parallel, meaning they never intersect or share a common value.

Two rival dry cleaners both advertise their prices. Let x equal the number of items dry cleaned. Store A's prices are represented by the expression 15x - 2. Store B's prices are represented by the expression 3(5x + 7). When do the two stores charge the same rate? Explain.

This problem involves comparing the prices of two dry cleaners using linear equations. Students will learn how to set up and solve an equation to determine if the prices of two services are ever equal. Ideal for students in Grades 8-10.

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