To solve this problem, we can use a system of linear equations. Let:

• $x$ be the amount invested in the 8% account.
• $y$ be the amount invested in the 5% account.

The total investment is $10,000$, so: $x + y = 10,000$

The total interest earned at the end of the year is $10,710 - 10,000 = 710$, so: $0.08x + 0.05y = 710$

We now have the system of equations:

1. $x + y = 10,000$
2. $0.08x + 0.05y = 710$

Step 1: Solve one equation for one variable

From the first equation: $y = 10,000 - x$

Step 2: Substitute into the second equation

Substitute $y = 10,000 - x$ into $0.08x + 0.05y = 710$: $0.08x + 0.05(10,000 - x) = 710$

Distribute the $0.05$: $0.08x + 500 - 0.05x = 710$

Simplify: $0.03x + 500 = 710$

Step 3: Solve for $x$

$0.03x = 710 - 500$ $0.03x = 210$ $x = \frac{210}{0.03} = 7,000$

Step 4: Solve for $y$

From $y = 10,000 - x$: $y = 10,000 - 7,000 = 3,000$

Final Answer:

• $x = 7,000$: $7,000 was invested in the 8% account.
• $y = 3,000$: $3,000 was invested in the 5% account.

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Let me know if you'd like further details or explanations! Here are some follow-up questions for deeper understanding:

1. How would the answer change if the interest rates were swapped?
2. What would happen if the total interest earned was different (e.g., $800)?
3. How can you verify the solution using substitution?
4. What if the total investment was not $10,000 but $12,000?
5. Can this problem be solved graphically?

Tip: When solving systems of equations, always check the solution by substituting back into the original equations.