Certainly! Let's go through the process of simplifying the algebraic expression (3a + 6ab - 6a - 2ab + 4a - 7c + 9b + (-1)b) step by step in detail.Step 1: Understanding the ExpressionThe expression consists of various terms involving different variables: (a), (b), and (c). Each term can be classified into three categories based on its structure:Constant terms: These do not appear in this specific expression.Linear terms: Terms that contain just one variable, such as (a) and (b).Product terms: Terms that contain products of variables, such as (ab).Step 2: Identifying Like TermsTo simplify the expression, we need to group and combine like terms. Like terms are those that have the same variable components.Terms involving (a):(3a)(-6a)(4a)Terms involving (ab):(6ab)(-2ab)Terms involving (b):(9b)((-1)b) (which can also be written as (-b))Terms involving (c):(-7c)Step 3: Combining Like TermsNow, we will combine the like terms identified in the previous step.Combining (a) TermsFor the terms involving (a):[ 3a - 6a + 4a ]Calculating this step by step:(3a - 6a = -3a)(-3a + 4a = 1a)So, the combined result for (a) terms is:[ a ]Combining (ab) TermsNext, for the terms involving (ab):[ 6ab - 2ab ]Calculating this gives:(6ab - 2ab = 4ab)Thus, the combined result for (ab) terms is:[ 4ab ]Combining (b) TermsNow, for the terms involving (b):[ 9b + (-1)b = 9b - b ]Calculating this gives:(9b - 1b = 8b)So, the combined result for (b) terms is:[ 8b ]Combining (c) TermsFinally, we have the term involving (c):[ -7c ]Since this term is standalone, it remains:[ -7c ]Step 4: Writing the Final ExpressionNow that we have combined all like terms, we can write the final simplified expression by combining our results:[ a + 4ab + 8b - 7c ]Step 5: ConclusionThe process of simplifying algebraic expressions involves identifying like terms, combining them through addition or subtraction, and ensuring that each type of variable is correctly accounted for. By following these steps carefully, we can simplify complex expressions systematically.Thus, the final simplified expression is:[ \boxed{a + 4ab + 8b - 7c} ]This expression is now in its simplest form, clearly presenting the relationship between the variables.