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An algebraic equation is a mathematical statement that asserts the equality of two expressions and generally includes variables, numbers, and arithmetic operations. For example, the equation from the exercise, \(y=(a+b)x-8\), features a combination of these elements. It includes variables \(x\) and \(y\), the constants \(a\), \(b\), and \(8\), and uses multiplication and subtraction.

In school, students are often tasked with solving algebraic equations for a specific variable, which entails performing operations to 'unlock' the value of that variable. Algebraic equations are foundational in mathematics because they represent relationships between quantities and allow for the prediction of unknown values. Understanding the principles behind these equations is crucial in higher levels of math, physics, economics, and many other fields.

Isolating a variable means rearranging an algebraic equation so that the variable of interest is on one side of the equal sign and everything else is on the other side. This manipulation simplifies calculating the value of the variable. In our example, we are isolating \(x\).

To achieve this, we perform inverse operations that balance the equation. In the exercise, we reverse the initial subtraction of \(8\) by adding \(8\) to both sides, getting us one step closer to isolating \(x\). It's essential to maintain the balance of the equation, meaning whatever operation is done to one side must be done to the other. After this step, we are left with \(y+8 = (a+b)x\), with \(x\) almost isolated.

Rearranging equations is a pivotal part of algebra that involves the reorganization of an equation to solve for a variable. This process requires an understanding of the inverse relationships between operations. After adding \(8\) to both sides of our initial equation to balance it, the next step in rearranging is to divide by \(a+b\) to completely isolate \(x\).

The division is necessary because \(x\) is originally being multiplied by \(a+b\). Using the inverse operation—division—we are left with \(x=\frac{y+8}{a+b}\), thus isolating \(x\). The ability to rearrange equations empowers students to solve for any variable, a skill that will be repeatedly used across many math-related subjects and real-life applications.