Problem 113 - College Algebra with Modeling and Visualization

Question

In \(2000,\) the combined population of Minneapolis/St. Paul, Minnesota, was \(670,000\). The population of Minneapolis was \(96,000\) greater than the population of St. Paul. (a) Write a system of equations whose solution gives the population of each city in thousands. (b) Solve the system of equations. (c) Is your system consistent or inconsistent? If it is consistent, state whether the equations are dependent or independent.

Step-by-Step Solution

Answer

Minneapolis: 383, St. Paul: 287; Consistent and independent system.

Step 1: Define the Variables

Let \(M\) represent the population of Minneapolis in thousands, and \(S\) represent the population of St. Paul in thousands.

Step 2: Create Equations Based on Given Information

From the problem statement, we have two pieces of information: the total population and the difference between the two populations. Therefore, we can write two equations. 1. The total population: \(M + S = 670\) 2. The population difference: \(M = S + 96\).

Step 3: Substitute to Solve for One Variable

Substitute the second equation into the first. Replace \(M\) in the first equation: \((S + 96) + S = 670\).

Step 4: Simplify the Equation

Combine like terms and simplify the equation from Step 3: \(2S + 96 = 670\).

Step 5: Solve for St. Paul's Population (S)

Subtract 96 from both sides to solve for \(S\): \(2S = 574\).Then, divide both sides by 2: \(S = 287\).

Step 6: Find Minneapolis' Population (M)

Using the value of \(S\) in the second equation: \(M = 287 + 96\).So, \(M = 383\).

Step 7: Check the Solution

Verify that the values satisfy both equations: 1. Total: \(383 + 287 = 670\), which is correct.2. Difference: \(383 = 287 + 96\), which is also correct.

Step 8: Analyze Consistency and Dependency

The system is consistent because it has a solution. The equations are independent since they provide two distinct constraints that are not scalar multiples of each other.

Key Concepts

Consistent and Inconsistent Systems

In the study of systems of equations, it's important to determine if a system is consistent or inconsistent. A **consistent** system is one that has at least one solution. That means the equations meet at a common point or share a solution set.
In the case of two linear equations such as

• The total population: \(M + S = 670\)
• The population difference: \(M = S + 96\)

the system is consistent because there is a distinct solution for the populations of Minneapolis and St. Paul.
An **inconsistent** system, on the other hand, does not have any solutions. This typically occurs when two equations represent parallel lines that never intersect. There is no point that satisfies both equations at the same time.

Dependent and Independent Equations

Determining if equations are independent or dependent is vital in analyzing the nature of the solutions. **Independent** equations offer distinct information. In other words, they constraint the variables in unique ways. For the solved system:

• The equation \(M + S = 670\) provides a total sum of the two populations.
• The equation \(M = S + 96\) suggests Minneapolis has a specific population difference from St. Paul.

These equations are independent because one equation cannot be derived from the other by simply multiplying by a non-zero constant. They provide two separate constraints that help to uniquely determine the solution.
In contrast, **dependent** equations are essentially the same line when graphed. They do not provide different constraints, meaning one equation is a simple transformation of the other and would not help solve the system uniquely unless both are used together with extra information.

Algebraic Problem Solving

Algebra is an essential tool for solving problems involving systems of equations. The process usually involves a series of logical steps:1. **Define Variables:** Start by establishing what each variable represents. Here, \(M\) and \(S\) represent the populations of Minneapolis and St. Paul, in thousands, respectively.
2. **Create Equations:** Translate the problem into mathematical expressions based on the information given. We have: - \(M + S = 670\) (sum of populations) - \(M = S + 96\) (difference in populations)
3. **Substitute to Simplify:** Combine these equations using substitution to eliminate one variable, making it simpler to solve. - Substitute \(M = S + 96\) into \(M + S = 670\), leading to a single-variable equation: \(2S + 96 = 670\).
4. **Solve for Variables:** Focus on one variable at a time to find concrete values. - Solving \(2S + 96 = 670\) gives \(S = 287\). - Use this to find \(M = 383\).

• This is the point where both initial equations are satisfied, providing the required solution.

Algebraic problem solving not only helps in finding solutions but also in verifying their accuracy through logical reasoning and systematic checking.