You invested a total of 8,400 in shares of the three stocks...

You invested a total of $\$ 8,400$ in shares of the three stocks at the given prices and expected to earn $\$ 248$ in annual dividends. If you purchased a total of 200 shares, how many shares of each stock did you purchase?

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Key Concepts

System of Linear Equations

This concept involves setting up and solving multiple linear equations that represent relationships among various unknown quantities. In problems involving investments or any scenario with multiple constraints, forming a system of equations allows for a structured way to isolate and determine each variable by ensuring that all conditions (such as totals and proportions) are satisfied concurrently.

Mathematical Modeling

Mathematical modeling is the process of translating a real-world situation into mathematical expressions. It involves identifying the relevant quantities, determining how they interact, and expressing these relationships using equations. This technique is widely used in finance, economics, and many other fields to analyze and solve practical problems based on given constraints.

Integer Constraints

In many scenarios, especially those involving counts or discrete items, the solution to an equation must be a whole number. Recognizing and applying these constraints is crucial, as it limits the solution set to integers only. This is commonly seen in problems where quantities, such as the number of shares or objects, cannot reasonably be fractional.

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