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When solving linear equations, one crucial step is combining like terms. This helps in simplifying the equation and making it easier to solve. Like terms have the same variables raised to the same power. In our original exercise, the terms \(-3.6x\) and \(8.2x\) on the left side of the equation are like terms because they both contain the variable \(x\). To combine them, simply add or subtract their coefficients (the numbers in front of the variables). Here, you calculate \(8.2 - 3.6\) to get \(4.6x\). Combining like terms reduces the equation to \(4.6x + 5.5 = 6.5 + 4.6x\), making it more straightforward to handle in subsequent steps. Remember, combining like terms means dealing with only their numerical parts, and keeping the variable part unchanged.

Once you have simplified an equation by combining like terms, the next step is to analyze it. You'll want to manipulate the equation so you can isolate the variable and solve it. In the simplified equation \(4.6x + 5.5 = 6.5 + 4.6x\), the variable terms \(4.6x\) are on both sides. Subtract \(4.6x\) from each side to try and isolate the variable. This subtraction is a common technique in equation analysis, often leading to seeing if like terms cancel out or simplify further. However, here you'll find all the variable terms cancel out, resulting in \(5.5 = 6.5\).This result is pivotal and helps in understanding the nature of the equation. It sets the stage for investigating deeper into what this equation really tells us.

When dealing with equations, you occasionally end up with a statement that has no logical solution. This occurs when the leftover equation doesn't make sense. In our case, after subtracting \(4.6x\) from both sides, we are left with \(5.5 = 6.5\).This is a classic example of a contradiction. It tells us that no value of \(x\) will ever make this equation true. A contradiction like \(5.5 = 6.5\) implies that the equation originally set up was inconsistent and inherently unsolvable.Recognizing a contradiction is essential because it tells you there is no feasible solution in the real number system. Rather than solving for \(x\), such an equation shows a deeper issue or misalignment in its formulation.