Problem 70 - College Algebra with Corequisite Support

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. An investor earned triple the profits of what she earned last year. If she made $$\$ 500,000.48$$ total for both years, how much did she earn in profits each year?

Step-by-Step Solution

Answer

The investor earned \$125,000.12 last year and \$375,000.36 this year.

Step 1: Define the Variables

Let the amount the investor earned the previous year be \( x \), and the amount earned this year be \( y \). We need to determine the values of \( x \) and \( y \).

Step 2: Set Up the Equations

According to the problem, the investor earned triple the profits this year compared to last year, which gives us the equation: \( y = 3x \). Additionally, the total earnings for both years are given as $$500,000.48, resulting in the equation: \( x + y = 500,000.48 \).

Step 3: Substitute the Expression

Substitute the expression for \( y \) from the first equation into the second equation: \( x + 3x = 500,000.48 \).

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

Combine like terms in the equation: \( 4x = 500,000.48 \). Solve for \( x \) by dividing both sides by 4: \( x = \frac{500,000.48}{4} = 125,000.12 \).

Step 5: Solve for \( y \)

Now that we have \( x = 125,000.12 \), substitute it back into the equation \( y = 3x \): \( y = 3 \times 125,000.12 = 375,000.36 \).

Step 6: Verify the Solution

Check that the total of \( x \) and \( y \) equals $500,000.48: \( 125,000.12 + 375,000.36 = 500,000.48 \). Verify that \( y = 3x \): \( 375,000.36 = 3 \times 125,000.12 \). Both conditions are satisfied.

Key Concepts

Variables and Equations

In algebra, a variable is a symbol that represents an unknown quantity. In solving problems, we often use variables to express relationships between different quantities. Equations, on the other hand, are mathematical statements that assert the equality of two expressions. For this type of problem, we commonly use two variables to represent unknown values.

For instance, in the exercise we looked at, the two variables are:

• Let \( x \) represent the amount the investor earned last year.
• Let \( y \) be the amount earned this year.

The relationships between these variables are represented by equations, which we solve to find their values. By setting up such variables and equations, we turn a real-world scenario into a mathematical model that can be manipulated to find unknown information.

Substitution Method

The substitution method is a way of solving a system of linear equations. It involves solving one of the equations for one variable and then substituting this expression into the other equation. This technique simplifies the system into one equation with one variable, making it easier to solve.

In our example:

• We first express \( y \) in terms of \( x \) using the equation: \( y = 3x \).
• Next, we substitute this expression into the second equation: \( x + y = 500,000.48 \), thus obtaining: \( x + 3x = 500,000.48 \).

This leaves us with a single equation \( 4x = 500,000.48 \) that can be solved to find \( x \) directly. The substitution method is powerful because it reduces the complexity of solving multiple variables.

Profit Analysis

In financial contexts, profit analysis enables us to understand how changes in one year’s profits impact earnings over time. It helps compare performances across different periods.

For this particular problem, profit analysis focuses on calculating and comparing the earnings from one year to the next:

• Last year’s earnings are captured by the variable \( x \), while this year’s profit is expressed as \( y = 3x \), illustrating that the investor tripled her profits.
• Identifying exact figures for each year allows for a clear analysis of performance improvement, such as \( x = 125,000.12 \) and \( y = 375,000.36 \), demonstrating the threefold increase.

A thorough profit analysis like this can inform future investment strategies and decisions as it offers precise financial insights.

Simplifying Equations

Simplifying equations is a crucial step in solving them efficiently. This means making an equation easier to handle by combining like terms or using basic arithmetic operations.

In our exercise:

• The expression \( x + 3x \) was simplified to \( 4x \).
• Subsequently, solving \( 4x = 500,000.48 \) involved basic division to isolate \( x \), giving: \( x = \frac{500,000.48}{4} = 125,000.12 \).

Simplifying equations often reduces complexity and avoids unnecessary calculations, making the process faster and lowering the potential for errors. It is a basic yet powerful tool in mathematics that enhances problem-solving efficiency.