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Linear equations are mathematical expressions where each term is either a constant or the product of a constant and a single variable. In this context, they are used to represent systems of equations where the relationships between variables are linear. These equations usually take on the standard form of \(ax + by + cz = d\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(x\), \(y\), \(z\) are variables.

This kind of equation describes straight lines in a geometric plane. Solving a system of linear equations involves finding the keys (or solutions) that satisfy all equations simultaneously.

• A typical system might include two or more equations.
• The objective is often to find values for the variables that make all the equations true at the same time.

Understanding linear equations is crucial for tackling more complex systems in mathematics.

Variable elimination is a method used to simplify and solve systems of equations. The goal is to remove one variable in order to allow for the solution of the others. This technique can simplify the process of finding solutions by focusing on fewer variables at a time.

In the original exercise, variable elimination was used to remove the variable \(z\) by combining equations. Here's how it works:

• Equations are manipulated algebraically (added or subtracted) to eliminate the variable.
• This creates a new equation with fewer variables, making it easier to solve.

Eliminating a variable helps reduce the system's complexity, breaking it down into smaller, more manageable sections.

The substitution method involves solving one equation for one variable and then substituting this expression into another equation. This process effectively reduces the number of variables in the equations, making it simpler to find solutions.

In our exercise, once \(z\) was expressed in terms of \(x\) and \(y\), this substitution made it possible to find terms that only included \(x\) and \(y\). This step is pivotal:

• Helps express variables in terms of others.
• Allows a gradual resolution of each variable one by one.

The substitution method is powerful because it makes solving complex systems manageable by isolating one variable at a time.

Solution verification is the process of checking whether the solutions found indeed satisfy the original equations. This step is crucial because it ensures that no mistakes were made during calculations.

To verify solutions, substitute the found values for variables back into the original equations:

• If all equations in the original system hold true, the solution is verified.
• Otherwise, errors might have occurred in prior calculations.

This step is essential in problem-solving as it confirms that the solutions are correct and valid across the system of equations.