The problem in the image asks to write and solve a system of equations based on the given simple interest situation:

1. Total investment is $13,000.
2. The interest generated is $500 per year.
3. The interest rates are 2% and 5%.

Define variables:

• Let $x$ be the amount invested at 2%.
• Let $y$ be the amount invested at 5%.

We know:

• $x + y = 13,000$ (the total amount invested is $13,000).
• The interest from each part is calculated using simple interest formula $I = P \times r$, where $P$ is the principal and $r$ is the rate.
• The total interest earned is $500. So, the interest equation is: $0.02x + 0.05y = 500$

System of equations:

1. $x + y = 13,000$
2. $0.02x + 0.05y = 500$

Solving the system:

Step 1: Solve for $y$ in terms of $x$ from the first equation:

$y = 13,000 - x$

Step 2: Substitute $y = 13,000 - x$ into the second equation:

$0.02x + 0.05(13,000 - x) = 500$ Expand the equation: $0.02x + 0.05 \times 13,000 - 0.05x = 500$ $0.02x + 650 - 0.05x = 500$ Combine like terms: $-0.03x + 650 = 500$ Solve for $x$: $-0.03x = 500 - 650$ $-0.03x = -150$ $x = \frac{-150}{-0.03} = 5,000$

Step 3: Solve for $y$:

$y = 13,000 - 5,000 = 8,000$

Conclusion:

• The amount invested at 2% is $5,000.
• The amount invested at 5% is $8,000.

Would you like further details or have any questions about this?

Here are 5 questions to deepen understanding:

1. How would the equations change if the total investment were different?
2. What happens if both interest rates were the same?
3. Can you use substitution or elimination to solve this system?
4. How does the simple interest formula change if the time is more than 1 year?
5. What if the total interest earned was $600 instead of $500?

Tip: Always check your solution by substituting the values back into both original equations to ensure they satisfy both conditions.