https://file.aiquickdraw.com/mathbot/file/17267524982005udywtlc.jpg

Let's break down the problem from the image step by step:

### Given:
- **Zahra's savings account**: Initial balance of \$450, and she saves \$50 each month.
- **Celine's savings account**: Initial balance of \$100, and she saves \$100 each month.

#### Part (a): Write a system of equations representing the scenario.

Let:
- $ t $ be the number of months.
- $ Z(t) $ represent Zahra's balance after $ t $ months.
- $ C(t) $ represent Celine's balance after $ t $ months.

The equations for both can be expressed as:

- For Zahra: 
  \[
  Z(t) = 450 + 50t
  \]
  (Initial balance of 450 plus 50 dollars per month)
  
- For Celine:
  \[
  C(t) = 100 + 100t
  \]
  (Initial balance of 100 plus 100 dollars per month)

#### Part (b): Graph the system of equations in Desmos.

To graph these, you would input the two equations into Desmos or any graphing tool:
1. $ Z(t) = 450 + 50t $
2. $ C(t) = 100 + 100t $

The intersection point of these two lines will show when Zahra and Celine have the same balance.

#### Part (c): Find when their balances are the same and how much they will have.

To find when their balances are equal, set the two equations equal to each other:
\[
450 + 50t = 100 + 100t
\]
Solve for $ t $:
\[
450 - 100 = 100t - 50t
\]
\[
350 = 50t
\]
\[
t = 7 \, \text{months}
\]

At $ t = 7 $, both will have the same balance. Plug $ t = 7 $ back into either equation to find how much they will have:

For Zahra:
\[
Z(7) = 450 + 50 \times 7 = 450 + 350 = 800
\]

For Celine:
\[
C(7) = 100 + 100 \times 7 = 100 + 700 = 800
\]

So, after 7 months, both Zahra and Celine will have \$800.

---

Would you like further details on any step? Here are five related questions:

1. How would the equations change if Celine saved \$120 instead of \$100 per month?
2. Can you verify the intersection point graphically using a tool like Desmos?
3. What would happen if Zahra saved \$10 more each month?
4. How would the graph look if Zahra started with a lower initial balance than Celine?
5. What if Zahra also saved in a different currency with an exchange rate applied monthly?

**Tip:** When solving systems of equations, graphing them is a powerful visual tool to understand their relationship, especially for real-world problems like this one.

Zahra opens a savings account with an initial balance of $450 and saves $50 each month. Celine opens a savings account with an initial balance of $100 and saves $100 each month. a) Write a system of equations representing this scenario. b) Graph the system of equations in Desmos. c) How many months will it take for Zahra and Celine to have the same balance? How much will they have at that point?

Explore how to model savings accounts for Zahra and Celine using a system of linear equations. Find out when their balances will be equal and graph the solution. This problem covers key concepts of linear functions and systems of equations, suitable for Grades 7-9.

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