Problem 3 - Algebra 2

Question

Jarrod Wright has a total of \(\$ 5000\) in his savings account and in a certificate of deposit. His savings account earns 3.5\(\%\) interest annually. The certificate of deposit pays 5\(\%\) interest annually if the money is invested for one year. He calculates that his interest earnings for the year will be \(\$ 227.50\). Write a system of equations for the amount of money in each investment.

Step-by-Step Solution

Answer

The savings account is $1500, and the certificate of deposit is $3500.

Step 1: Define Variables

Let \( x \) be the amount of money in the savings account and \( y \) be the amount in the certificate of deposit. The total money in both accounts is \$5000.

Step 2: Set Up First Equation

The first equation comes from the total amount in both investments: \[ x + y = 5000 \] which represents that the sum of the savings account and the certificate of deposit is \$5000.

Step 3: Set Up Second Equation Using Interests

The second equation is based on the interest earned from both accounts: \[ 0.035x + 0.05y = 227.50 \] representing the total interest earned in a year from both accounts.

Step 4: Solve the System of Equations

Use substitution or elimination methods to solve the equations. We'll use substitution:- From the first equation, express \( y \) in terms of \( x \): \[ y = 5000 - x \]- Substitute \( y \) in the second equation: \[ 0.035x + 0.05(5000 - x) = 227.50 \]- Simplify and solve for \( x \): \[ 0.035x + 250 - 0.05x = 227.50 \] \[ -0.015x + 250 = 227.50 \] \[ -0.015x = -22.50 \] \[ x = 1500 \]

Step 5: Find the Value of y

With \( x = 1500 \), substitute back to find \( y \):\[ y = 5000 - 1500 = 3500 \]

Key Concepts

Interest Calculation

Calculating interest can be approached with simplicity if you understand the basic principle: it's all about understanding how much extra money you make from an investment over time. In Jarrod's situation, he has two forms of investment: a savings account and a certificate of deposit. Each type pays interest at a different rate.

The savings account earns interest at an annual rate of 3.5%, meaning if you have a balance of \\(100, you'll earn \\)(100 \times 0.035) = \\(3.50 over one year.

For the certificate of deposit, the interest rate is 5%. With the same \\)100, you earn \\((100 \times 0.05) = \\)5.00 annually. These calculations show how different interest rates affect earnings, and this concept forms the foundation for interest-related financial decisions.

Jarrod's total interest earnings are given as \$227.50 per year, so he combines the interest from both accounts to match this total with his investments.

Substitution Method

The substitution method is a classic algebraic technique used to solve systems of equations. It's about replacing one variable with an expression from another equation, which simplifies the process of solving.

In Jarrod's problem, the first equation is formed by the total amount in each investment: \([x + y = 5000]\). The second is based on total interest, \([0.035x + 0.05y = 227.50]\).

First, you solve the first equation for one variable, saying, \(y = 5000 - x\). Substitute this expression for \(y\) into the second equation. You then work with a single equation in terms of just \(x\):

• Start by expanding and simplifying: \(0.035x + 0.05(5000 - x) = 227.50\).
• Continue simplifying: \(-0.015x + 250 = 227.50\).
• Solve for \(x\): \(-0.015x = -22.50\), leading us to \(x = 1500\).

This method helps break down complex systems into simpler, solvable parts, making it a highly efficient and reliable method for solving linear equations.

Elimination Method

The elimination method is another effective strategy for tackling systems of equations. This method involves adding or subtracting equations to eliminate one variable, simplifying the path to the solution.

While the substitution method was used in Jarrod's example, elimination could also solve the problem. Here's a quick look at how it could work:

• Take both equations: \(x + y = 5000\) and \(0.035x + 0.05y = 227.50\).
• Multiply the first equation by 0.035 to set up elimination: \(0.035x + 0.035y = 175\).
• Subtract this from the second equation to eliminate \(x\):\(0.015y = 52.50\).
• Solve to find \(y\): \(y = 3500\).
• Use this \(y\) in the first equation to find \(x\).

This approach emphasizes aligning equations such that direct subtraction or addition ends up removing one variable, often streamlining the path to a solution.

Investment Distribution

Investment distribution involves deciding how to allocate funds across various investment options to maximize returns. In this exercise, Jarrod Wright divides his \\(5000 between a savings account and a certificate of deposit.

The challenge is to find how much was invested in each account, given that the total interest earnings are \\)227.50.

By solving his system of equations using either substitution or elimination, we discover Jarrod placed \\(1500 into the savings account and \\)3500 in the certificate of deposit.

Knowing how to distribute funds wisely is crucial in personal finance, especially when different investments yield different interest rates. By understanding each option's rate, you can effectively spread your investment and reach financial goals faster. In essence, this exercise not only teaches solving equations but imparts fundamental financial literacy lessons that benefit long-term planning.