Larry Mitchell invested part of his $\$ 19,000$ advance at $5 \%$ annual simple interest and the rest at $2 \%$ annual simple interest. If his total yearly interest from both accounts was $\$ 560$, find the amount invested at each rate.
The amount invested at $5 \%$ is $\$ \square$.
The amount invested at $2 \%$ is $\$ \square$.
Answer
\(\boxed{\text{The amount invested at 5% is }\$ 6000\text{. The amount invested at 2% is }\$ 13000\text{.}}\)
Step 1 ：Let the amount invested at 5% be x and the amount invested at 2% be y. We have two equations:
Step 2 ：1. \(x + y = 19000\) (total investment)
Step 3 ：2. \(0.05x + 0.02y = 560\) (total interest)
Step 4 ：Solve this system of linear equations to find the values of x and y.
Step 5 ：Solution: \(x = 6000\) and \(y = 13000\)
Step 6 ：\(\boxed{\text{The amount invested at 5% is }\$ 6000\text{. The amount invested at 2% is }\$ 13000\text{.}}\)
