https://file.aiquickdraw.com/mathbot/file/17267056216971k3wy15c.jpg

The problem provided asks us to determine after how many weeks Lucy and Emma will have saved the same amount of money, given their initial savings and weekly contributions.

### Step 1: Identify given values.
- **Lucy** starts with **$10** and saves **$5** weekly.
- **Emma** starts with **$15** and saves **$4** weekly.

### Step 2: Formulate the equation.
Let **$ x $** represent the number of weeks after which they will have saved the same amount. The total savings for each person after $ x $ weeks can be represented as:

- Lucy's savings: $ 10 + 5x $
- Emma's savings: $ 15 + 4x $

### Step 3: Set up the equation.
Since we want to know when Lucy's and Emma's total savings will be equal:

\[
10 + 5x = 15 + 4x
\]

### Step 4: Solve the equation.
We now solve for $ x $.

\[
10 + 5x = 15 + 4x
\]
Subtract $ 4x $ from both sides:

\[
10 + x = 15
\]
Subtract 10 from both sides:

\[
x = 5
\]

### Step 5: Conclusion.
Lucy and Emma will have the same amount of money after **5 weeks**.

The equation that represents this situation is:  
\[
10 + 5x = 15 + 4x
\]
The solution is:  
They will have the same amount of money after **5 weeks**.

---

Would you like more details on any of the steps or have any other questions?  
Here are 5 related questions you might find interesting:
1. How would the equation change if Lucy saved $6 weekly instead of $5?
2. What if Emma started with $20 instead of $15, how many weeks would it take then?
3. Can you graph the savings for both Lucy and Emma to show where they intersect?
4. How would the solution change if both Lucy and Emma had the same initial amount of money but different savings rates?
5. What if Lucy decided to stop saving after 3 weeks, how would that affect the solution?

**Tip:** When forming equations, always double-check that the expressions for both sides correctly represent the situation described!

Lucy and Emma plan to pool and save their allowance for a new video game they want to play together. Both want to contribute an equal amount of money. Lucy already has $10 and plans to save $5 weekly. Emma has $15 and plans to save $4 each week. After how many weeks will they have saved the same amount of money?

Explore the problem of saving money over time with Lucy and Emma. Learn to solve linear equations such as 10 + 5x = 15 + 4x to find when both will have the same amount of money. This problem is suitable for students in Grades 6-8 and uses basic algebraic principles.

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