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Understanding the technique of grouping like terms is pivotal when solving linear equations. It involves collecting the variables (often represented as 'x' or 'y') on one side of the equation and the constant numbers on the other. This separation streamlines the process of isolating the variable, bringing you one step closer to finding its value.

For instance, in the equation \(2x - 7 = 6 + x\), we can move the terms containing 'x' to one side by subtracting 'x' from both sides. Similarly, we can move the constant numbers to the opposite side by adding 7 to both sides. This rearrangement is crucial because it simplifies the equation and sets the stage for the next step: simplification.

Once the like terms are grouped together, the next step in solving a linear equation is simplifying the equation. Simplification may sound complex, but it's essentially a process of performing arithmetic operations to reduce the equation to its simplest form, where the variable stands alone with a clear coefficient.

Continuing from our grouping exercise, the equation \(2x - x = 6 + 7\) appears after rearrangement. Simplification involves combining the 'x' terms (resulting in a single 'x') and adding the constants together (resulting in 13), leading to an equation as straightforward as \(x = 13\). This stripped-down version makes the solution clear and easy to understand, showcasing the power of simplification.

To ensure that our solution is correct, checking the solution is a fail-safe step. By substituting the variable with our supposed solution in the original equation, we can verify its accuracy.

For the equation \(2x - 7 = 6 + x\), if we replace 'x' with 13, we get \(2(13) - 7 = 6 + 13\). This simplifies to \(26 - 7 = 19\) on the left side and \(19\) on the right side. Since both sides equal 19, the original equation holds true, confirming that \(x = 13\) is indeed the correct solution. This checking step is essential as it provides confidence in the solution and solidifies the understanding of the problem-solving process.