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Solving for a variable in an algebraic equation is like finding the key to a puzzle. In algebra, isolating the variable means rearranging the equation so that the variable you're solving for is alone on one side of the equality sign.

To isolate a variable, you perform a series of operations that will 'undo' anything that's being done to the variable. For example, consider the equation \( -x + 5 = 0 \). The variable \( x \) is subtracted from 5, so to isolate \( x \) you would do the opposite: add \( x \) to both sides, resulting in \( 5 = x \).

Remember, whatever you do to one side of the equation, you must do to the other to keep the equation balanced. This way, the action of isolating the variable doesn't change the equation's meaning or its solution.

Once you believe you've found the solution to an equation, verifying your answer ensures that it truly works. To verify the solution, insert the value back into the original equation and check if the equation remains in balance.

In the context of our example, after isolating and finding \( x = 5 \), we can verify it by plugging 5 back into the original equation, giving us \( -5 + 5 \). This simplifies to \( 0 \), which matches the right side of the equation \( -x + 5 = 0 \). Thus, the solution \( x = 5 \) is confirmed to be accurate. It's like making sure the key not only fits in the lock but also turns to open it.

Algebraic expressions are combinations of numbers, variables, and operations. They are the building blocks of algebra and form the sentences of a mathematical language. For instance, in the equation \( -x + 5 = 0 \), the sequence \( -x + 5 \) is an expression consisting of a variable \( x \) and constants.To unravel the meanings behind these expressions, understanding the fundamentals is crucial—subtraction, addition, multiplication, division, and the use of negative numbers all contribute to forming these expressions. They are not just random assemblies of symbols; each part of an expression plays a role in the equation's overall story.