Problem 59 - Algebra A Combined Function

Question

The number of personal bankruptcy petitions filed in the United States was consistently on the rise until there was a major change in bankruptcy law. The year 2007 was the year in which the fewest personal bankruptcy petitions were filed in 15 years, but the rate soon began to rise. In \(2009,\) the number of petitions filed was 206,593 less than twice the number of petitions filed in 2007 . This is equivalent to an increase of 568,751 petitions filed from 2007 to 2009. Find how many personal bankruptcy petitions were filed in each year. (Source: Based on data from the Administrative Office of the United States Courts)

Step-by-Step Solution

Answer

775,344 petitions in 2007 and 1,344,095 petitions in 2009.

Step 1: Understanding the Problem

We need to find the number of personal bankruptcy petitions filed in 2007 and 2009. There are two key pieces of information: (1) the number of petitions in 2009 was 206,593 less than twice the number in 2007, (2) the number in 2009 was 568,751 more than in 2007.

Step 2: Setting Up the Equations

Let's denote the number of petitions in 2007 as \( x \). According to the problem, the number of petitions in 2009 is 206,593 less than twice the number in 2007. Therefore, we can represent the 2009 petitions as \( 2x - 206,593 \). We also know that the same number is 568,751 more than the number in 2007, so it can also be expressed as \( x + 568,751 \).

Step 3: Solving the Equations

Now we have two expressions for the number of petitions in 2009: \( 2x - 206,593 \) and \( x + 568,751 \). We set these equal to each other: \[2x - 206,593 = x + 568,751\] Subtract \( x \) from both sides to get: \[x - 206,593 = 568,751\] Add 206,593 to both sides:\[x = 568,751 + 206,593 = 775,344\] So, 775,344 petitions were filed in 2007.

Step 4: Finding the 2009 Petitions

Now that we have \( x = 775,344 \), we can substitute it back into the expression for 2009 petitions: \\[2x - 206,593 = 2(775,344) - 206,593\]\[= 1,550,688 - 206,593 = 1,344,095\] Thus, 1,344,095 petitions were filed in 2009.

Key Concepts

Linear Equations

Linear equations are fundamental tools in algebra and crucial for solving real-world problems. A linear equation is an equation that represents a straight line when plotted on a graph. It generally has the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable that we solve for. These equations can be easily manipulated to find the value of the unknown variable.

In our problem, the key lies in setting up and solving a linear equation that helps us find the number of bankruptcy petitions filed in 2007. We use the given relationships between the numbers from different years to form equations. By comparing different expressions for the number of petitions filed, we eventually solve for \(x\), which denotes the petitions filed in 2007.

Here are some steps involved in solving linear equations:

• Identify the variable and constants.
• Use given relationships to set up the equation.
• Simplify the equation by performing operations such as addition, subtraction, multiplication, or division.
• Solve for the variable to find its value.

Understanding and solving linear equations are crucial skills in algebra, paving the way for effective problem-solving.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (such as addition and subtraction) that represent values and relationships in mathematical terms. They serve as the building blocks for setting up equations and models.

In the exercise, the algebraic expressions \(2x - 206,593\) and \(x + 568,751\) were crucial in describing the relationship between the number of petitions filed in 2007 and 2009. These expressions capture the constraints provided in the problem and allow us to establish an equation.

Some important features of algebraic expressions include:

• Variables: Represent unknown values, such as \(x\) in our problems.
• Constants: Known values added, subtracted, or multiplied by variables (like 206,593 and 568,751).
• Operators: Symbols such as +, -, *, and / that denote operations that need to be applied to elements in the expression.

By manipulating these expressions, we can isolate the variable and find its value straightforwardly, making algebraic expressions an essential tool in problem-solving.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions and equations to represent real-world situations. This process provides a structured approach to solving practical problems by translating them into mathematical language.

In our exercise, we used mathematical modeling to understand the increase in bankruptcy petitions. By defining \(x\) as the number of petitions filed in 2007, we created a model that describes the changes from 2007 to 2009 using linear equations and algebraic expressions. This model enables us to analyze the relationship between different variables and find specific numeric values.

Key steps in mathematical modeling include:

• Defining the variables to represent real-life quantifiable aspects.
• Establishing relationships and constraints using mathematical language.
• Solving the resulting equations to answer specific questions about the situation.
• Interpreting the results back into the context of the original problem.

Mathematical modeling is a powerful technique that allows students and researchers to understand and solve complex problems using precise mathematical tools.