Problem 54 - College Algebra with Corequisite Support

Question

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. You invest \(\$ 80,000\) into two accounts, \(\$ 22,000\) in one account, and \(\$ 58,000\) in the other account. At the end of one year, assuming simple interest, you have earned \(\$ 2,470\) in interest. The second account receives half a percent less than twice the interest on the first account. What are the interest rates for your accounts?

Step-by-Step Solution

Answer

The interest rate for the first account is 2%, and for the second account is 3.5%.

Step 1: Define Variables

Let's define variables for the interest rates of the two accounts. Let \( r_1 \) be the interest rate for the first account, and \( r_2 \) be the interest rate for the second account.

Step 2: Set Up the Equations

We know that the total interest earned from both accounts is $2,470. This gives us the first equation: 1. \( 22000r_1 + 58000r_2 = 2470 \).According to the problem statement, the interest rate for the second account is half a percent less than twice the interest rate of the first account. This gives us the second equation: 2. \( r_2 = 2r_1 - 0.005 \).

Step 3: Substitute Second Equation into First Equation

Substitute the second equation into the first equation to eliminate \( r_2 \) from the system:\[ 22000r_1 + 58000(2r_1 - 0.005) = 2470 \].Simplify the equation to find \( r_1 \).

Step 4: Simplify and Solve for \( r_1 \)

First, distribute 58000 in the equation:\[ 22000r_1 + 116000r_1 - 290 = 2470 \].Combine like terms:\[ 138000r_1 - 290 = 2470 \].Add 290 to both sides:\[ 138000r_1 = 2760 \].Divide both sides by 138000 to solve for \( r_1 \) :\[ r_1 = \frac{2760}{138000} \approx 0.02 \].

Step 5: Solve for \( r_2 \) Using \( r_1 \)

Using the value found for \( r_1 \) in the second equation to find \( r_2 \):\[ r_2 = 2(0.02) - 0.005 = 0.04 - 0.005 = 0.035 \].

Step 6: Verify the Solution

To verify, substitute \( r_1 = 0.02 \) and \( r_2 = 0.035 \) back into the first equation:\[ 22000(0.02) + 58000(0.035) = 440 + 2030 = 2470 \].The equation matches the given total interest, confirming our solution is correct.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations with multiple variables. Each equation represents a line, and the solution is the point(s) where these lines intersect. In our exercise, we deal with two equations involving the interest rates of two accounts, represented by variables \( r_1 \) and \( r_2 \). These variables correspond to the interest rates of two different investment accounts.

We set up these equations based on the total interest earned and the relationship between the two interest rates. The first equation captures the total interest from both accounts, while the second defines the relationship between the rates. To solve this system using Cramer's Rule, we need to express it in terms of matrices. This approach allows us to find solutions by calculating determinants, providing a straightforward way to solve the equations once set up.

Interest Rates

Interest rates represent the cost of borrowing money or the reward for saving. It's often expressed as a percentage. In our problem, we focus on interest rates for investments, which determine how much interest we earn annually.

The rates are vital because they directly impact the total return on an investment. Here, we have two accounts with interest rates \( r_1 \) and \( r_2 \). Understanding how these rates contribute to the total interest earned is crucial to solving the problem. This task involves finding exact rates that satisfy both the total interest requirement and the specific condition given for \( r_2 \).

Knowing how to set and manipulate these rates within equations is an essential skill in finance and algebra.

Simple Interest

Simple interest is a way to calculate the interest amount based on the original investment, or principal. Unlike compound interest, which accrues on both the initial principal and the accumulated interest, simple interest grows linearly.

The formula for simple interest is:

• \( \,\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \, \)

In this problem, we only consider one year, simplifying our calculations.

By applying this formula to both accounts, we create expressions for the interest earned, which we then use to form our system of linear equations. Understanding this straightforward method is particularly useful for introductory finance or algebra courses.

Variables

Variables are symbols used to represent unknown values in equations. They allow us to create general formulas and solve for unknowns. In our mathematical context, \( r_1 \) and \( r_2 \) are variables that stand for the unknown interest rates of the accounts.

By defining these variables, we convert words and relationships into mathematical expressions. This simplification makes it easier to solve complex problems systematically.

In any financial situation involving multiple factors, identifying and manipulating variables is crucial. It transforms abstract concepts into concrete numbers, enabling us to analyze and solve real-world problems efficiently.