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In algebra, expression simplification is the process of making an expression easier to understand or work with. It involves performing operations such as addition, subtraction, and distributing any negative signs or coefficients.
When simplifying \((6-j)-9-(2j-3)\), we begin by addressing the most straightforward operation first: distributing the negative sign. This means each element within the parentheses must be altered by the negative. Consequently, \((6-j)\) remains unchanged, but \(-(2j-3)\) becomes\(-2j + 3\). By expanding and removing parentheses, we translate the expression to:

• Positive and negative adjustments: \(6-j-9-2j+3\)

This step is essential as it sets the stage for further simplification, where we can focus on grouping and combining like terms for a cleaner expression.

Combining like terms is a crucial step in the simplification process of algebraic expressions. It involves adding or subtracting terms that have the same variable component and power. For instance, in algebraic expressions, terms with the same variable like \(j\) must be combined together.
In our expression \(6-j-9-2j+3\), we focus on two types of terms:

• Constants: \(6\), \(-9\), and \(+3\)
• Variable terms: \(-j\) and \(-2j\)

Combine constants first: \(6 - 9 + 3\) results in \(0\). This step reduces numerical parts to a manageable form. Next, focus on variables. Combining \(-j\) and \(-2j\) results in \(-3j\). Therefore, the expression simplifies to \(-3j\).
Combining like terms ensures the algebraic expression is simplified efficiently and is easier to handle in future calculations.

Algebraic verification is about confirming the reliability of your simplified expression by using a specific value for the variables involved. It's a test to verify the accuracy of the simplification. This process is instrumental in spotting possible errors.
Let's take \(-3j\), the simplified expression, and verify it by substituting \(j\) with \(1\). For the original expression:

• \((6-1)-9-(2\times1-3) = 5 - 9 - (-1) = 5 - 9 + 1 = -3\)

Now, for the simplified expression: \(-3j\) substituted as \(-3\times1\), results in \(-3\).
Both the original and simplified expressions return the result \(-3\), confirming the accuracy of the simplification. This practice of verification helps reinforce confidence in algebraic procedures and results.