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Understanding the distributive property is crucial when solving linear equations involving parentheses. In the equation \[-2(x+1) + 3x = 14,\]we have to handle the expression inside the parenthesis first. The distributive property allows us to "spread" the multiplication of a number over addition or subtraction inside the parentheses.
This means that you multiply \(-2\) by \(x\) and \(1\), giving us the terms \(-2x\) and \(-2\). After applying the distributive property, the equation becomes:\[-2x - 2 + 3x = 14.\]
Using the distributive property correctly helps simplify equations, making it easier to analyze and solve them.

Once you've simplified the equation using the distributive property, the next step involves "combining like terms". Like terms are terms in an equation that have the same variable raised to the same power. For example, in \(-2x - 2 + 3x = 14,\) the like terms on the left side are \(-2x\) and \(3x\).
By combining these, we simplify the equation:\[-2x + 3x = x.\]This step reduces the number of terms and transforms the equation into a simpler form, leading us to:\[x - 2 = 14.\]
Combining like terms is a foundational skill in algebra that helps to keep equations manageable and sets the stage for isolating the variable.

The final goal in solving a linear equation is to isolate the variable, which is the unknown component you are solving for. In our newly simplified equation \[x - 2 = 14,\]we need to get \(x\) by itself on one side of the equation.
To do this, you perform the opposite operation of what's being done to \(x\). Here, \(x\) is being subtracted by \(2\), so we add \(2\) to both sides:\[x - 2 + 2 = 14 + 2.\]This cancels out the \(-2\) on the left side, leaving:\[x = 16.\]Remember, the goal of isolating the variable is to end up with something like \(x = ext{number}\), effectively solving for \(x\). In more complex equations, this might involve several steps, but the concept remains the same.