Problem 106 - College Algebra with Modeling and Visualization

Question

Investments \(\quad\) A total of \(\$ 5000\) was invested in two accounts. One pays \(5 \%\) annual interest, and the second pays \(7 \%\) annual interest. If the first-year interest is \(\$ 325\), how much was invested in each account?

Step-by-Step Solution

Answer

\$1250 in the 5% account and \$3750 in the 7% account.

Step 1: Define Variables

Let \( x \) be the amount of money invested in the account with 5% interest, and \( y \) be the amount of money invested in the account with 7% interest.

Step 2: Write the Equation for Total Investment

The total investment is \$5000. Therefore, we have the equation:\[ x + y = 5000 \]

Step 3: Write the Equation for Total Interest

The total interest for the first year is \$325. The interest from the first account is \(0.05x\) and from the second account is \(0.07y\). Thus, the equation is:\[ 0.05x + 0.07y = 325 \]

Step 4: Solve the System of Equations

We now have two equations:1. \( x + y = 5000 \)2. \( 0.05x + 0.07y = 325 \)We can use substitution or elimination to solve for \(x\) and \(y\). We'll use substitution. From the first equation, express \(y\) in terms of \(x\):\[ y = 5000 - x \]Substitute \(y\) in the second equation:\[ 0.05x + 0.07(5000 - x) = 325 \]Simplifying:\[ 0.05x + 350 - 0.07x = 325 \]\[ -0.02x + 350 = 325 \]\[ -0.02x = -25 \]\[ x = \frac{-25}{-0.02} = 1250 \]

Step 5: Find the Second Variable

Now that we have \(x = 1250\), substitute it back into the equation \(y = 5000 - x\):\[ y = 5000 - 1250 = 3750 \]

Step 6: Verify the Solution

Verify the solution by checking if the calculated interests add up to \$325:- Interest from the first account: \(0.05 \times 1250 = 62.5\)- Interest from the second account: \(0.07 \times 3750 = 262.5\)- Total interest: \(62.5 + 262.5 = 325\)The interests match, verifying that the solution is correct.

Key Concepts

Linear Equations

Linear equations are a fundamental component of algebra and are used extensively to solve a variety of real-world problems. A linear equation is any equation that can be written in the form \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants.

These equations are characterized by variable terms (e.g., \( x \) and \( y \) in the exercise) combined using addition, subtraction, and multiplication, but not division of variables or powers higher than one.

Solving linear equations typically involves finding the value of the variable that makes the equation true.

• To solve a system of linear equations like the one in the exercise, you can use methods such as substitution, elimination, or graphically representing the equations to find points of intersection.

These methods allow you to determine the value of unknowns that satisfy all given equations simultaneously.

Investment Problems

Investment problems often involve finding the allocation of amounts or resources to achieve a desired financial goal.

In these problems, various accounts or investments may offer different returns or interest rates, and the challenge lies in balancing different allocations to optimize the outcome—such as maximizing returns or meeting a specific requirement.

Incorporating equations is crucial in dealing with investment problems.

• Setting up the right equations helps in organizing the given information effectively.
• It allows one to compute unknown variables such as amounts invested in each account, interest earned, or overall profits.

This type of problem can often be solved by establishing a system of equations, just like we did in the exercise, where we balanced the total amount invested and the interest generated to find the answer.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to solve problems by representing real-world situations in abstract terms. It involves creating equations or systems of equations that reflect a problem's conditions and constraints.

Modeling is crucial because it provides a structured path to finding solutions. In the given exercise, we used mathematical modeling by defining variables and forming equations to model the investment scenario.

• This involves identifying what quantities are unknown and what relationships exist between known and unknown quantities.
• It also requires formulating equations based on those relationships, which describe how elements interact, guiding you to a resolution.

Understanding the concept of mathematical modeling empowers you to apply mathematics effectively to diverse fields beyond simple exercises, such as optimizing business processes, predicting stock markets, or improving engineering designs.