Problem 54 - College Algebra

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 rates are 2% for the first account and 3.5% for the second account.

Step 1: Identify Variables

Let \( r_1 \) be the interest rate in decimal form for the first account, and \( r_2 \) for the second account. These are the values we need to find.

Step 2: Set Up Equations

From the problem, we have two accounts with interest amounts totaling to \$2,470. Therefore, one equation is: \[ 22,000r_1 + 58,000r_2 = 2,470. \] Additionally, the interest rate on the second account \( r_2 \) is half a percent less than twice the interest rate of the first account, giving us:\[ r_2 = 2r_1 - 0.005. \]

Step 3: Substitute and Simplify

Substitute \( r_2 = 2r_1 - 0.005 \) into the first equation:\[ 22,000r_1 + 58,000(2r_1 - 0.005) = 2,470. \] Simplify this to:\[ 22,000r_1 + 116,000r_1 - 290 = 2,470. \] Combine like terms to get:\[ 138,000r_1 = 2,760. \]

Step 4: Solve for First Account Rate

Solve for \( r_1 \) by dividing both sides of the equation by 138,000:\[ r_1 = \frac{2,760}{138,000} = 0.02. \] So, the interest rate for the first account is 2%.

Step 5: Determine Second Account Rate

Use the equation \( r_2 = 2r_1 - 0.005 \) to find \( r_2 \):\[ r_2 = 2(0.02) - 0.005 = 0.035. \] Thus, the interest rate for the second account is 3.5%.

Key Concepts

System of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. In the given exercise, we had two variables: the interest rates for two accounts, represented as \( r_1 \) and \( r_2 \). These variables help us create equations that describe the financial situation. By setting up two equations based on the information given:

• One equation for the total interest earned by both accounts.
• Another for the relationship between the two interest rates.

We then solve these equations simultaneously. By using Cramer's Rule or substitution, we can find the unique solution for \( r_1 \) and \( r_2 \). This reflects how important systems of linear equations are in solving real-world financial problems, such as determining interest rates.

Interest Rates

Interest rates determine how much additional money you earn or pay on an investment or loan over a period of time. Expressed as a percentage, they can significantly impact your financial returns. In this problem, our goal was to find the interest rates for two accounts after one year.The first account, invested at an interest rate \( r_1 \), and the second, at \( r_2 \), needed to sum to a total of \$2,470 in interest. Subtle connections between the rates were described: the rate of the second account was less straightforward, being half a percent less than twice the rate of the first account. This complexity highlights how interest rates provide both opportunities and variables to manage when investing.

Simple Interest

Simple interest is calculated using a straightforward formula: \( \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \). In this exercise, we assumed the situations applied to simple interest. The calculations for each account followed this formula, and the interest was direct and predictable.

• The first account generated interest based on \\(22,000 at rate \( r_1 \).
• The second account involved \\)58,000 at rate \( r_2 \).

At the end of the year, calculating simple interest allowed for a direct assessment of how the principal interacted with interest rates over time, equating to a tangible benefit for the investor.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and systems of linear equations. It's crucial for solving problems where there are multiple variables and equations, like in our investment scenario. By representing the interest rates as variables, linear algebra helps us solve for these unknowns with efficiency. In the exercise, significant aspects of linear algebra included:

• Creating matrices from systems of equations.
• Utilizing methods such as substitution or Cramer's Rule to find solutions.
• Understanding how matrix operations can provide clear insight to otherwise complex problems.

Through linear algebra, calculating the interest rates becomes systematic and manageable, opening the door to deeper financial analysis and decision-making.