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Linear equations are an integral part of algebra and are the starting point for solving a variety of real-world problems. These equations describe a straight line when plotted on a graph and typically take the form of \( ax + b = 0 \), where \(a \) and \(b\) are constants, and \(x\) is the variable we aim to solve for.

When faced with a linear equation, the primary goal is to isolate the variable on one side of the equation, leaving a constant on the other side. This is achieved through a series of steps that might include distributing, combining like terms, and using inverse operations to simplify the equation. The process ensures that in the end, we get \(x\) to stand alone with a numerical value that represents its solution.

Equation simplification is the process of making an algebraic equation easier to understand and solve. This could involve a variety of strategies such as combining like terms, which are terms that have the same variable raised to the same power, or canceling out terms on both sides of the equation to get closer to the solution.

For example, if we have an equation like \( 8a + 2 = 6a + 8 \), we can subtract \(6a\) from both sides to simplify it to \(2a + 2 = 8\). Next, we can subtract \(2\) from both sides resulting in \(2a = 6\), and finally divide by \(2\) to get \(a = 3\). Such steps are the essence of equation simplification and are universally applicable to solve for the variable in focus.

The distributive property is a cornerstone of algebra which allows us to multiply a single term by each term inside a bracket (parenthesis) in an expression. It states that \( a(b + c) = ab + ac \) where \(a\), \(b\), and \(c\) could be any number, variable, or a combination of both.

This property is highly useful when dealing with equation simplification as it helps to eliminate brackets and pave the way for further manipulation of the equation. Applied incorrectly, however, it can lead to errors that throw off the solution. Therefore, it is crucial to distribute the terms carefully and correctly. For instance, in the provided exercise, a mistake was made either in the distribution of terms or in the operations that followed, resulting in an incorrect check. A correct application of the distributive property would have helped in accurately simplifying the equation for the correct solution.