Problem 72 - College Algebra

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. If an investor invests a total of \(\$ 25,000\) into two bonds, one that pays 3\(\%\) simple interest, and the other that pays 2\(\frac{7}{8} \%\) interest, and the investor earns \(\$ 737.50\) annual interest, how much was invested in each account?

Step-by-Step Solution

Answer

The investor invested $15,000 in the 3% bond and $10,000 in the 2 7/8% bond.

Step 1: Define Variables

Let \( x \) be the amount invested in the bond with 3\(\%\) interest and \( y \) be the amount invested in the bond with 2\(\frac{7}{8}\%\) interest. We want to find the values of \( x \) and \( y \).

Step 2: Set Up Equations

We have two pieces of information. First, the total amount invested is \\(25,000, which gives us the first equation: \[ x + y = 25000 \]Second, the total interest earned from both investments is \\)737.50. The interest from the first bond is \(0.03x\) and from the second bond is \(\frac{23}{8y}\), leading to the second equation: \[ 0.03x + \frac{23}{800}y = 737.5 \]

Step 3: Simplify the System of Equations

Simplify the interest equation by multiplying through by 800 to clear the fraction:\[ 24x + 23y = 590000 \] Now the system of equations is:1. \( x + y = 25000 \)2. \( 24x + 23y = 590000 \)

Step 4: Solve the System of Equations by Substitution

Express \( y \) from the first equation:\[ y = 25000 - x \]Substitute \( y \) in the second equation:\[ 24x + 23(25000 - x) = 590000 \]This simplifies to:\[ 24x + 575000 - 23x = 590000 \]\[ x = 15000 \]

Step 5: Solve for the Second Variable

Substitute \( x = 15000 \) back into the expression for \( y \):\[ y = 25000 - 15000 \]\[ y = 10000 \]

Step 6: Verify the Solution

Check if the values satisfy the second equation (interest equation):\[ 24(15000) + 23(10000) = 360000 + 230000 = 590000 \]Both the total investment and interest equations are satisfied, confirming the solution is correct.

Key Concepts

Linear Equations with Two Variables

When dealing with a system of linear equations with two variables, we typically have two main equations that interrelate two unknowns. For example, in our exercise, these unknowns are the amounts invested in two different bonds. These equations allow us to solve real-world problems like this by finding an optimal value for these variables.

• Each equation represents a constraint on possible solutions.
• By combining and solving these equations, we can pinpoint exact mystery values, like investment amounts in this scenario.

One common method is setting one equation equal to a variable and plugging it into the other equation. This method helps break down the problem into manageable parts. Through substitution, we make complex systems less intimidating and solve for each unknown step by step.

Simple Interest

Simple interest is a way to calculate the interest on loans or investments. It relies purely on the initial amount of money, or principal, without building on the previous interest. This makes it distinct from compound interest, which does consider accumulated interest in its calculations.

• Interest formulas generally use: \( I = P \cdot r \cdot t \), where \( I \) is interest, \( P \) is principal, \( r \) is rate, and \( t \) is time in years.
• Our exercise uses simple interest, where the investor receives a fixed percentage, based on the principal, annually.

By understanding simple interest, we gain insight into straightforward finance calculations, such as understanding how much earnings an investment generates in a year's span.

Bond Investment Problems

Bond investment problems involve deciphering the amounts allotted to different investments that generate various returns. In these problems, some bonds pay higher interest rates than others. Solving such problems demand an understanding of different interest rates and their application to specific portions of invested capital.

• For bonds, different investment portions result in different interest earnings.
• Balancing total investment with individual bond returns can unravel what amount goes where.

By formulating equations based on these constraints, like in our example, we can precisely determine investments in each bond, maximizing returns and strategically planning financial goals.

Solving Equations by Substitution

Solving equations by substitution is a practical way to tackle problems involving multiple variables. It involves expressing one variable in terms of another, substituting into the second equation, and simplifying.

• Start by isolating a variable in one equation.
• Substitute this expression into the other equation, reducing the system to a single variable.

For our specific problem, we expressed one bond investment as the total minus the second bond investment. Plugging it back, we simplified our equations, ultimately solving for one variable before quickly finding the second. This method is essential for breaking down complex problems, ensuring an orderly approach to finding solutions.