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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.

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.

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.