91Ó°ÊÓ

Algebraic manipulation refers to the ability to rearrange and simplify algebraic equations or expressions using various rules and properties of arithmetic operations. It's a foundational skill in algebra that enables learners to transform complex problems into more solvable forms.

When solving the equation \(x-1=cx+d\), the first step involves algebraic manipulation to isolate the terms containing \(x\). By subtracting \(cx\) from both sides of the equation, we're using the property of equality – what's done to one side must be done to the other to maintain balance. This manipulation simplifies the problem into a form where the variable \(x\) can be more easily isolated and solved.

Factoring is a critical process in algebra where we express an algebraic expression as a product of its factors. Factors can be numbers, variables, or expressions. It's most often used as a precursor to further simplification, especially when solving equations.

In our equation, once we have \(x(1-c)-1=d\), we identify \(x\) as a common factor in the terms \(x\) and \(-cx\). Factoring \(x\) out from these terms, we end up with the expression \((1 - c)x - 1 = d\). This is an important step, as it organizes the equation into a clearer form where \(x\) appears only once, which consequently eases the process of isolating the variable.

The isolation of variables is a goal-oriented strategy where we manipulate an equation to ultimately express a single variable on one side of the equal sign, and all the other terms on the opposite side. This process gives us the solution to the variable in terms of other constants or variables present in the equation.

After factoring, we have the equation \((1 - c)x = d + 1\). To isolate \(x\), we need to eliminate all terms attached to \(x\) by executing inverse operations. Adding \(1\) to both sides removes the subtraction of 1 from the \(x\)-term. Following that, we divide both sides by the coefficient of \(x\), which is \((1-c)\). The final result is a clear separation: \(x = \frac{d+1}{1-c}\), thus achieving the isolation of the variable \(x\).