Problem 73 - College Algebra

Question

Use a system of linear equations with two variables and two equations to solve. If an investor invests $$\$ 23,000$$ into two bonds, one that pays \(4 \%\) in simple interest, and the other paying \(2 \%\) simple interest, and the investor earns $$\$ 710.00$$ annual interest, how much was invested in each account?

Step-by-Step Solution

Answer

\( x = 12,000, y = 11,000 \)

Step 1: Define Variables

Let \( x \) represent the amount invested at \( 4\% \) interest and \( y \) represent the amount invested at \( 2\% \). We know the total investment is $23,000, so we can write this as an equation.

Step 2: Write System of Equations

From the problem's conditions, we have two equations:1. \( x + y = 23000 \) (total investment)2. \( 0.04x + 0.02y = 710 \) (total interest)

Key Concepts

Linear Equations

Linear equations are mathematical sentences that express relationships between variables using equality signs. These equations typically contain one or more variables, each raised to the power of one, which is why we call them 'linear.' In our exercise, the equations involve the variables \( x \) and \( y \), which are used to solve the problem of investment distribution. Linear equations can be solved using various methods, such as graphing, substitution, or elimination.
In the given exercise, we use linear equations to determine how \( \$23,000 \) was split between two different bonds. By formulating equations based on given conditions, we precisely capture how these investments relate to each other under fixed constraints, such as total investment and total accumulated interest. This format makes linear equations incredibly useful for solving a variety of real-life problems, including financial investments.

Investment Calculation

Investment calculation is crucial in determining how funds are allocated among different investment options to maximize returns. This involves identifying the different components of investment and typical regulations affecting it. To solve these types of problems, a clear understanding of initial investments, individual returns, and overall yields is required.
In this exercise, we set up our problem by letting the amount invested at \( 4\% \) be \( x \) and the amount at \( 2\% \) be \( y \). Together, these amounts total \( \$23,000 \). Breaking down investments like this allows us to calculate returns more straightforwardly and manage financial expectations effectively. Through careful mathematical modeling, investors can predict their earnings based on different strategies.

Simple Interest

Simple interest is a way to calculate the interest charged on an initial principal over a certain period of time without compounding. It is commonly utilized in bonds, loans, and investments to determine the straightforward return on investment. The formula for simple interest is \( I = P \times r \times t \), where \( I \) is the interest, \( P \) is the principal amount, \( r \) is the rate of interest, and \( t \) is the time period.
In our exercise, simple interest helps us calculate the total earnings from two bonds. One bond yields interest at \( 4\% \), and another at \( 2\% \). By setting up the equation \( 0.04x + 0.02y = 710 \), we determine the annual earnings from the investments. The use of simple interest in such problems is straightforward, as it provides a predictable pattern of earnings, assisting investors in planning and decision-making.