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Combining like terms is one of the fundamental steps when solving linear equations. This concept involves adding or subtracting terms that have the same variable part.
In the exercise: \(4x + 3 + 8x + 1 = 11x + 2\), you'll notice several terms with the variable \(x\) and constant terms without any variables.

The first step is to add the coefficients (numbers in front) of the terms with \(x\). In this case, we combine \(4x\) and \(8x\), resulting in \(12x\) (since \(4 + 8 = 12\)). The same rule applies to constants. Here, \(3 + 1\) gives \(4\).
Combining these terms simplifies the equation to: \(12x + 4 = 11x + 2\).

By mastering combining like terms, you can simplify complex-looking equations into more manageable parts.

Isolating the variable is an essential step in solving linear equations. The goal is to get the variable term alone on one side of the equation.
From our simplified equation: \(12x + 4 = 11x + 2\), we want all \(x\) terms on one side and constants on the other.

Start by getting rid of the \(11x\) term on the right. Subtract \(11x\) from both sides: \(12x + 4 - 11x = 11x + 2 - 11x\)

This step simplifies our equation to: \(x + 4 = 2\), where we successfully isolated the variable on one side.
The next task will involve solving this simpler equation.

Solving for \(x\) is the final step where we find the value of the variable that makes the equation true. With the variable isolated, our equation is now: \(x + 4 = 2\).

To solve for \(x\), we need to get rid of the \(4\). We do this by subtracting \(4\) from both sides: \(x + 4 - 4 = 2 - 4\).

After performing this operation, we get: \(x = -2\).

Solution verification is crucial: substitute \(-2\) for \(x\) in the original equation to ensure the left and right sides are equal.
Understanding each step helps make solving linear equations straightforward and less intimidating.