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Linear equations are a central part of solving systems of equations, such as the one given in the hiking problem. A linear equation typically has one or more variables, and all the terms of the equation are either constants or products of constants and single variables. Let's break this down:- **Standard Form**: Linear equations commonly appear in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants and \( x \) and \( y \) are variables. Each equation represents a straight line when graphed on a coordinate plane.- **Characteristics**: Linear equations do not have variables raised to any power higher than one, no products of variables, and no changes in the degree of equations.- **Relevance in Systems of Equations**: In a system of linear equations, we look for values of the variables that satisfy all equations simultaneously. In the given problem, the equations \( x + y = 4,824 \) and \( x - y = 476 \) are both linear equations expressed in terms of the lengths of the trails.

When tackling a system of equations, defining your variables is a critical first step. Variables serve as placeholders for the unknowns you are trying to find. Here's how you can effectively define variables:- **Purpose**: Defining variables helps translate a real-world problem into mathematical language, making it easier to solve. Once defined, these variables are used throughout the solution process.- **Approach**: Choose variable names that are intuitive or use the problem context for naming. For instance, in the hiking problem, the variables could be defined as: - \( x \) for the length of the Pacific Crest Trail - \( y \) for the length of the Appalachian Trail- **Usage**: With these definitions, you can systematically create equations based on the conditions given in the problem statement, paving the way for finding solutions.

Solving methods are techniques used to find the solution to a system of equations. There are several methods, each useful in different scenarios. Here, we explore the addition method, commonly used for its simplicity:- **Addition Method (or Elimination)**: This method involves adding or subtracting equations to eliminate one variable, thereby simplifying the solution process. - **Example**: Given the equations \( x + y = 4,824 \) and \( x - y = 476 \), adding them eliminates \( y \), resulting in \( 2x = 5,300 \), which simplifies to \( x = 2,650 \). - **Substitution**: After finding one variable, substitute back to find the other. Once \( x \) is known, substitute into the equation \( x + y = 4,824 \) to solve for \( y \), leading to \( y = 2,174 \).- **Choice of Method**: Other methods include substitution and graphing, but the addition method is often preferred for linear systems due to its straightforward process and effectiveness in reducing the number of terms quickly.