https://file.aiquickdraw.com/mathbot/file/1725537755727m2yewn1m.jpg

Let's break down the problem in the image step by step.

### Given Information:
- Jack's grandfather gave a check to start a college savings account.
- They deposited **$175 per month** after opening the account.
- After **5 years**, there is **$13,000** in the account.
- This amount **does not include interest**.

### Part a) Write an equation to model the account balance, $ y $, after $ x $ years.

We know that:
- Each year has 12 months, so the monthly deposit of $175 translates to an annual deposit of $ 175 \times 12 = 2100 $ dollars per year.
- Let the initial check amount from Jack's grandfather be $ C $.
- The total account balance after $ x $ years would be the initial amount $ C $ plus the deposits made over $ x $ years.

The equation to represent the balance after $ x $ years is:
\[
y = C + 2100x
\]

### Part b) What was the check's original amount?

We are told that after 5 years, the balance is $13,000. Using the equation from part a), we can substitute $ y = 13000 $ and $ x = 5 $ to find the value of $ C $.

\[
13000 = C + 2100 \times 5
\]
\[
13000 = C + 10500
\]
\[
C = 13000 - 10500
\]
\[
C = 2500
\]

So, the original check amount from Jack's grandfather was **$2,500**.

---

### Summary of Answers:
a) The equation for the account balance after $ x $ years is:
\[
y = C + 2100x
\]

b) The original check amount was **$2,500**.

---

Would you like more details on any part of the solution? Here are some related questions to explore further:

1. What if the monthly deposit was increased to $200? How would the equation change?
2. How would interest affect the model if it were added at 2% annually?
3. What would be the balance after 10 years, assuming the same monthly deposit rate?
4. How would you modify the model if Jack's parents skipped one year of monthly deposits?
5. If the grandfather gave a different initial amount, how would that change the equation and final balance?

**Tip:** Always carefully check units (e.g., monthly vs. annual amounts) when modeling financial problems.

Jack's grandfather gave his parents a check to start a college savings account. They opened the account with the check, and then deposited $175 per month since. After 5 years, there is $13,000 in the account. Write an equation to model the account balance after x years and find the original check amount.

Solve a problem involving a college savings account where $175 is deposited monthly, and the balance after 5 years is $13,000. Learn how to write a linear equation and find the original check amount from Jack's grandfather.

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