Problem 70 - College Algebra

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

Last year's profit was $125,000.12, and this year's profit was $375,000.36.

Step 1: Define Variables for Profits

Let's begin by defining variables for each year's profits. Let \( x \) represent the profits from last year, and \( y \) represent the profits from this year.

Step 2: Set Up Equations from Problem Statement

According to the problem, the earnings this year are triple those of last year, giving us the equation \( y = 3x \). Moreover, the total earnings over the two years are \( \$ 500,000.48 \), giving us the equation \( x + y = 500000.48 \).

Step 3: Substitute and Solve for One Variable

Substitute \( y = 3x \) into the second equation: \( x + 3x = 500000.48 \). This simplifies to \( 4x = 500000.48 \).

Step 4: Solve for Last Year's Profits

Divide both sides of the equation \( 4x = 500000.48 \) by 4 to find \( x \): \( x = \frac{500000.48}{4} = 125000.12 \).

Step 5: Solve for This Year's Profits

Substitute \( x = 125000.12 \) back into the equation \( y = 3x \) to find \( y \): \( y = 3 \times 125000.12 = 375000.36 \).

Key Concepts

Two Variables

In this exercise, we deal with a system of linear equations that involve two variables. These variables represent unknown quantities that we want to determine. Here, we define:

• \( x \) as the profits earned last year
• \( y \) as the profits earned this year

By assigning these variables, we can translate the problem statement into equations. For instance, recognizing that this year’s profits are triple that of last year helps us form a critical equation: \( y = 3x \).
This equation relates the two unknowns, making it possible to solve them simultaneously. This step of defining variables is crucial as it simplifies complex real-world statements into manageable mathematical expressions. The goal is always to decipher and represent the given information using equations involving the two variables.

Substitution Method

The substitution method is a powerful technique for solving systems of linear equations. It involves solving one of the equations for one variable and then substituting this expression into the other equation.
In our exercise, we first express \( y \) in terms of \( x \) from the equation \( y = 3x \).By substituting \( 3x \) in place of \( y \) in \( x + y = 500000.48 \), we simplify the system. This approach reduces the number of variables you work with in the equation to just one, instantly making it simpler.

• Start by expressing a variable from one equation.
• Substitute the expression into the other equation.
• Solve the resulting equation.

This technique is particularly useful because it leads directly to an equation with one variable, streamlining the path to a solution.

Problem Solving

Problem solving with systems of linear equations involves breaking down a situation into mathematical expressions. The key is to interpret a real-world scenario logically and translate it into a system of equations. In this exercise, we:

• Identify and assign variables to unknowns.
• Develop equations based on known relationships.
• Apply a method, like substitution, to find solutions.

A systemic approach helps in navigating complex problems, where each step builds on the previous one. By focusing on variable relationships and total quantities, we reduce ambiguity.
Accurate interpretation and representation of information are what drive successful problem solving.

Algebraic Equations

Algebraic equations form the foundation for expressing relationships in a system of equations. In our exercise, these equations derive from understanding the problem:

• Relation between this year's and last year's profits: \( y = 3x \)
• Total profits over two years: \( x + y = 500000.48 \)

Each equation captures a specific detail of the scenario, illustrated through a mathematical relationship established through variables. Solving algebraic equations requires:

• Manipulating terms to isolate variables.
• Balancing equations through operations like addition, subtraction, multiplication, or division.

Mastering these techniques is essential for uncovering unknowns in any real-world scenario defined by algebraic equations.

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