Problem 73 - College Algebra with Corequisite Support

Question

For the following exercises, 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

\$12,500 was invested at \(4\%\) and \$10,500 at \(2\%\).

Step 1: Define the variables

Let's define:- Let \( x \) be the amount invested at \(4\%\).- Let \( y \) be the amount invested at \(2\%\).

Step 2: Set up the total investment equation

Since the total investment is \(\$23,000\), we have:\[ x + y = 23,000 \]

Step 3: Set up the interest equation

The total interest earned is \(\$710\). The interest from the \(4\%\) bond is \(0.04x\) and from the \(2\%\) bond is \(0.02y\). Thus, the interest equation is:\[ 0.04x + 0.02y = 710 \]

Step 4: Solve the equations by substitution

First, solve the first equation for \(y\):\[ y = 23,000 - x \]Substitute \(y\) in the interest equation:\[ 0.04x + 0.02(23,000 - x) = 710 \]Simplify and solve for \(x\):\[ 0.04x + 460 - 0.02x = 710 \]\[ 0.02x = 250 \]\[ x = 12,500 \]

Step 5: Find the value of y

Now, substitute \(x\) back into the equation for \(y\):\[ y = 23,000 - 12,500 \]\[ y = 10,500 \]

Step 6: Verify the solution

Check the total interest calculation:- Interest from \(\\(12,500\) at \(4\%\) is \(0.04 \times 12,500 = 500\).- Interest from \(\\)10,500\) at \(2\%\) is \(0.02 \times 10,500 = 210\).- Total interest is \(500 + 210 = 710\), which matches the given total.

Key Concepts

Simple Interest

Simple interest is a method used to calculate the interest charged or earned on an investment or loan. It is determined by multiplying the principal amount by the interest rate and the time period of the investment or loan. With simple interest, the interest is not compounded, meaning you do not earn interest on the interest. This is unlike compound interest, where the interest itself earns interest over time.

• Principal: The initial amount of money invested or borrowed.
• Interest rate: The percentage of the principal that is paid as interest over a specific period.
• Time: The duration over which the interest is calculated.

For example, if you invest \(1,000 at a simple interest rate of 5% for one year, you would earn \)50 in interest. This calculation uses the formula: \( \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \), without adapting the principal amount for any generated interest.

Linear Algebra

Linear algebra is a branch of mathematics that deals with the study of vectors, vector spaces, and linear equations. It is fundamental in understanding systems of linear equations, which involve multiple linear equations involving the same set of variables. In this math problem, linear algebra allows us to model and solve the problem using a system of equations.

• Linear Equation: An equation that forms a straight line when graphed. It has the general form: \( ax + by = c \).
• System of Linear Equations: A collection of two or more linear equations involving the same variables. They express constraints that must all be satisfied at the same time.

Linear algebra helps us find solutions for such systems, often using methods like substitution or elimination, ensuring that all constraints are respected simultaneously. It is useful for investment problems where multiple conditions need to be solved together.

Investment Problem

An investment problem often involves determining how to allocate money among multiple options to achieve a financial goal. In mathematical terms, it entails setting up equations based on investments and the returns they produce. This specific investment problem is about dividing $23,000 between two bonds with different interest rates to reach a total annual interest of $710. The steps to solve an investment problem include:

• Identifying the investment options and their characteristics, such as interest rates and total sum available for investment.
• Creating equations based on the total investments and expected returns.
• Solving these equations using algebraic methods to find how much money is allocated to each investment option.

This allows an investor to effectively strategize how much to place in different accounts while working within the constraints of expected returns.

Variables and Equations

In math and problem-solving, variables and equations are crucial concepts. A variable is a symbol used to represent a number in equations and formulas, and when dealing with two or more variables, you'll often create a system of equations to solve a problem.In our exercise, the variables are:

• \( x \): representing the amount invested at 4% interest.
• \( y \): representing the amount invested at 2% interest.

Equations are mathematical statements expressing the equality between different expressions. They are essential for forming the foundation to solve real-world problems like our investment exercise. By creating equations that reflect the conditions of the problem, we can use algebra to find the values of these variables. This allows us to understand and calculate the solution to how much was invested in each option, ensuring that all criteria are met effectively.