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Linear equations are mathematical expressions that represent straight lines when plotted on a graph. They are called 'linear' because they describe a straight-line relationship between two variables. A general form of a linear equation in two variables is given by the equation:

• \( ax + by = c \)

Here \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. These equations are crucial because they are the simplest form of algebraic expressions and the building blocks for understanding more complex equations. Linear equations are often used in solving real-world problems, from calculating finances to engineering tasks.

In the original exercise, the linear equations derived from parametric equations are:

• \( x_2 - x_1 = 1 \)
• \( x_1 + x_3 = 2 \)

These equations help stipulate the relationship between the variables \( x_1 \), \( x_2 \), and \( x_3 \). They highlight how linear equations can derive from and work with parametric equations like \( x_1 = t \), conveying solutions that are both practical and theoretical in nature.

A system of equations is a set of two or more equations that have common variables. The primary goal is to find values for these variables that satisfy all the equations simultaneously. Solving systems of equations is a foundational skill in algebra. It is used to determine where two or more lines intersect, or in other words, where their values satisfy all conditions given by the equations.

For example, in the context of the exercise, the system is represented as:

• \( x_2 - x_1 = 1 \)
• \( x_1 + x_3 = 2 \)

These have to be satisfied simultaneously, meaning there's a specific set of \( x_1, x_2, \) and \( x_3 \) that will satisfy both equations.

Solving such systems involves various techniques like substitution, elimination, or graphing to find potential solutions. In the given solution, substitution is used to express \( x_2 \) and \( x_3 \) in terms of \( x_1 \), reducing the system to easily manageable terms. The system is coherent and solvable for any parameter \( t \) or newly defined parameter \( s \), showcasing the flexibility of systems of equations in different forms.

Variables are symbols used to represent unknown values or values that can change. In mathematics and equations, they are typically denoted by letters such as \( x \), \( y \), \( t \), or \( s \). These symbols stand in for numbers that either need to be found or can vary within a certain context.

In our exercise, the variables \( x_1, x_2, \) and \( x_3 \) are represented in terms of the parameter \( t \). These variables can take on infinite sets of values depending on \( t \). For example:

• \( x_1 = t \)
• \( x_2 = 1 + t \)
• \( x_3 = 2 - t \)

The exercise introduces flexibility by switching the parameter from \( t \) to \( s \), which affects how the variables are expressed and their derived relationships through parametric and system equations.

Understanding how variables operate within different mathematical structures, such as parametric equations and systems, helps students analyze problems from multiple perspectives and apply creativity to find solutions. It's essential to grasp how variables interact, transform, and represent multiple facets of a problem.