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The distributive property is a cornerstone in the realm of algebra. It's basically the rule that says when you have an expression that multiplies a number by the sum (or difference) of two other numbers, you can 'distribute' the multiplication to each individual term. This looks like:
For example, if we are given \( a(b + c) \), the distributive property lets us expand this to \( ab + ac \). This principle is also applicable in reverse; you can factor out a common term from several terms. In our textbook exercise, we see the distributive property in action when \( -4 \) is distributed to the terms inside the parentheses, changing \( -4(5 + x) \) into \( -20 - 4x \).
This property is vital for solving equations because it allows us to simplify expressions so that we can combine like terms next.

After utilizing the distributive property, the equation often has terms that can be combined, known as 'combining like terms'. This process is a bit like consolidating items; just as you might put all your socks in one drawer, you combine all like terms in an equation into a single term.
'Like terms' have the exact same variable raised to the same power. For example, \( 2x \) and \( 5x \) are like terms because they both have \( x \) to the first power. However, \( 2x \) and \( 5x^2 \) are NOT like terms because one is \( x \) while the other is \( x^2 \). In our exercise example, we combined \( 12x \) and \( -4x \) to get \( 8x \), simplifying the equation and setting the stage to isolate the variable next.

The final phase of solving an algebraic equation is 'isolating the variable'. This means manipulating the equation so that the variable you're solving for stands alone on one side of the equation, allowing you to see its value clearly.
In our problem, we want to solve for \( x \). Once like terms are combined, as we've done in the earlier step, we're ready to isolate \( x \). This involves performing operations that undo what's being done to \( x \). For instance, if \( x \) is being subtracted by 20, we do the opposite, which is adding 20, to both sides of the equation to cancel this term out. This leaves us with \( 8x = 24 \). The last step is to divide by the coefficient 8 to get \( x \) by itself, resulting in \( x = 3 \). Isolating variables is akin to peeling away layers to reveal the core of a problem, leading us to the solution.