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In the context of investment problems, a system of equations is a set of two or more mathematical statements expressing certain conditions or relationships. Each equation represents a different aspect of the problem, such as the total amount invested or the total interest received. Here, we have two primary conditions:

• The sum of the amounts invested at different interest rates is equal to the total investment.
• The total interest from these investments meets a specified goal.

Setting up these equations requires identifying what each variable represents. For example, in our problem:

• Let \( X \) be the amount invested at 7%.
• Let \( Y \) be the amount invested at 8%.

Using these variables, our system of equations becomes:

• \( X + Y = 15000 \)
• \( 0.07X + 0.08Y = 1100 \)

These equations highlight the problem's constraints and what needs solving. Grasping this concept is essential, as it sets the stage for employing a method, like substitution, to find the unknowns.

To solve a system of equations, the substitution method is a straightforward and effective approach. This method involves isolating one variable in one equation and then substituting this expression into the other equation. Let's break it down:First, pick one of the equations and isolate one of the variables. In our example, we start with the equation:\[ X + Y = 15000 \]We can solve for \( Y \):\[ Y = 15000 - X \]Next, substitute this expression for \( Y \) into the second equation:\[ 0.07X + 0.08(15000 - X) = 1100 \]By substituting, we reduce two equations into one equation with one variable. Solving this simplifies our task:

• Multiply: \( 0.07X + 1200 - 0.08X = 1100 \)
• Simplify: \( -0.01X + 1200 = 1100 \)
• Solve for \( X \): \( -0.01X = -100 \)
• \( X = 10000 \)

This method is beneficial in investment problems, as it aligns well with the structure of financial equations. It ensures a step-by-step isolation and simplification, making it easier to track and verify the solution.

Investment problems often involve determining how much money is invested at various rates of return. These problems are mathematical reflections of real-life financial scenarios. The key is understanding the core components:

• Amount of money invested in different interests.
• Rates of return – these are expressed as percentages, such as 7% or 8%.
• Total return or interest and ensuring it meets a specified criterion.

In our particular problem, the total investment is \(15,000, and the woman receives \)1,100 in interest annually. The challenge was to find out how much was invested at each rate. Real-life applications include rebalancing a portfolio or comparing different investment vehicles. By setting up a system of equations, applying the substitution method, and carefully validating the solution:

• We determine \( X = 10000 \) for the amount at 7%.
• This leads to \( Y = 5000 \) when we substitute back into \( Y = 15000 - X \).

Understanding these problems equips individuals with the ability to make informed financial decisions, ensuring they can optimize for their desired interest or risk, balancing out different investments for the best possible outcome.