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Investment problems often involve determining how money is distributed across different investment options. These can include stocks, bonds, mutual funds, or other investment vehicles. The goal is usually to figure out the amounts invested in each option based on a set of conditions and constraints.
For instance, you might be given the total amount of investment, different rates of return for each option, total returns, or specific relationships between the invested amounts.
Such problems can be effectively solved using a system of linear equations. This helps in simultaneously solving for multiple unknowns by using the given conditions as equations.
Understanding investment problems is crucial as it applies mathematical concepts to real-world financial decisions.

When solving investment problems, you often encounter multiple unknown values, or 'variables', that need to be determined.
In the example problem, we have three variables:

• \(x\) for the amount invested in stocks
• \(y\) for the amount invested in bonds
• \(z\) for the amount invested in the mutual fund

Each variable represents a specific segment of the total investment. Using three variables makes it possible to create a system of equations that models the entire problem.
With the variables defined, it becomes easier to set up equations based on the given parameters such as total investment, returns on investment, and relationships between different investments. This structured approach simplifies the problem-solving process significantly.

Solving a system of linear equations with three variables involves several steps. First, you set up three equations based on the problem's conditions.
For example, in the given problem, we use:

• Total investment: \( x + y + z = 25000 \)
• Total return: \( 0.08x + 0.10y + 0.06z = 1860 \)
• Relationship between investments: \( z = 2y \)

With these equations established, you can use substitution or elimination methods to find the values of the variables.
Substitution involves replacing one variable with an equivalent expression from another equation. You solve one of the equations in terms of one variable and then substitute this expression into the other equations.
Elimination involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the remaining variables.
Let's break down the solution:

• First, substitute \( z = 2y \) into the first and second equations
• Rearrange and solve one of the equations for a single variable, such as \( x \)
• Use the obtained value to find the other variables
• Finally, verify the solution by plugging the values back into the original equations

This structured method ensures that the equations are solved systematically and correctly.