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Linear equations are fundamental in algebra and deal with expressions where each term is either a constant or the product of a constant and a single variable. Simply put, linear equations represent straight lines when graphed on a coordinate plane. These equations are called "linear" because they form straight lines. They follow the basic form:
* **Standard form:** \( Ax + By = C \)* **Slope-intercept form:** \( y = mx + b \)* **Point-slope form:** \( y - y_1 = m(x - x_1) \)
In our problem, the system consists of three linear equations, involving three variables: \(x\), \(y\), and \(z\). Our goal is to find a common solution set \((x, y, z)\) that satisfies all these linear equations simultaneously.
The beauty of linear equations is their simplicity and predictability. When solving these systems algebraically, each equation might provide a piece of the puzzle. Together, they lead us to the complete solution.

To solve equations, especially in a system, mathematical operations are utilized to find the value of unknown variables that make the equation true. In this context, we deal with systems of linear equations, where multiple equations must hold simultaneously.
We often use methods like:

• Substitution: Solving one equation for a variable and then substituting this expression into another equation.
• Elimination: Adding or subtracting equations to eliminate one of the variables.
• Matrix methods: Utilizing row operations to solve equations, especially useful for larger systems.

In our solution: * The elimination method was initially used. By adding different pairs of equations, we eliminated variables systematically. * We transitioned to substitution by solving for one variable in terms of others and subsequently plugging it back. This structured approach eventually leads to the unraveling of each variable, thus solving the system.

When tackling systems of linear equations, a step by step solution is crucial to ensure clarity and precision. Each phase of the solution contributes crucially to understanding how the final answer is reached.
Here's a brief walkthrough of our exercise:

• Step 1: Clearly identify your equations. The clarity at this stage determines the efficiency of subsequent steps.
• Step 2: Look for opportunities to eliminate variables. This simplifies the system into fewer equations with fewer variables.
• Step 3: Simplify where possible. Reduced and simplified equations make the system more manageable.
• Step 4-5: Progressively isolate variables, substituting known values to resolve unknowns incrementally.
• Step 6-7: Continue isolating and substituting until each variable's value is identified.
• Step 8: Always verify your solution with the original equations to ensure correctness.

Each step carefully builds on the last, revealing the values of \(x\), \(y\), and \(z\) that satisfy the system. This methodical progression is what makes mathematics both logical and satisfying, transforming abstract symbols into concrete solutions.