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Understanding the distributive property is essential when simplifying algebraic expressions. It allows us to multiply a single term by each of the terms in a parenthesis-segregated expression.

Consider the expression from the exercise, the distributive property is used to 'distribute' the number outside the parenthesis, in this case, 2, to each term inside the parenthesis, here, 5x and -1. Mathematically, this is represented as:
\[ 2(5x - 1) = 2 \cdot 5x + 2 \cdot (-1) \]
By applying this property, we eliminate the parenthesis and transform the expression into a simpler form. Here's a breakdown:

• Multiply 2 by 5x, which equals 10x.
• Multiply 2 by -1, which gives -2.

So, after applying the distributive property, the expression becomes \(10x - 2\). Now, we can proceed to the next steps, knowing that we correctly distributed the multiplication.

After distributing, we often find ourselves with an expression that has similar terms—these are called 'like terms.' Like terms are terms that contain the same variable raised to the same power.

In our example, after distribution, we have \(10x - 2 + 14x\). Now, we identify like terms: \(10x\) and \(14x\) which both have the variable x to the first power. To combine them, we simply add their coefficients:
\[10 + 14 = 24\]
Thus, combining the like terms \(10x\) and \(14x\), we get \(24x\). The expression now reads \(24x - 2\). Combining like terms is like tidying up: it streamlines our expression, making it easier to understand and utilize in further calculations.

The final step in simplifying an algebraic expression is to bring together all the simplification skills, like distribution and combining like terms, to form the simplest version of the expression. Algebraic simplification might also include adding or subtracting constant terms, simplifying coefficients, or factoring.

In the given exercise, after distributing and combining like terms, the expression is already in its simplest form: \(24x - 2\). There are no further like terms to combine, and the terms that remain do not share a common factor that would allow us to factor the expression.

Algebraic simplification leads to an expression that is easier to work with, whether that means substituting in values for the variables or using the expression in larger equations. It's the crucial final touch in ensuring that you're working with an expression that's as clear and concise as possible.