91Ó°ÊÓ

An algebraic equation is a statement of equality between two expressions, composed of variables and constants. Algebraic equations are the cornerstone of algebra and provide a powerful tool for solving a wide range of problems.

For example, in the equation presented in the exercise, \(r-1=16\), \(r\) represents the variable, while \(1\) and \(16\) are constants. The challenge is to determine the value of the variable that makes the equation true. Understanding how to manipulate these equations to find the variable's value can open the door to solving more complex problems in mathematics and beyond.

The primary goal when solving linear equations is to isolate the variable - this means getting the variable by itself on one side of the equation while keeping the equation balanced. Isolating the variable is a fundamental skill in algebra, as it allows us to find the value of the variable that satisfies the equation.

In our example, the variable \(r\) is initially not by itself because it is accompanied by \(-1\). To isolate \(r\), the opposite of subtraction, which is addition, is used. By adding \(1\) to both sides, the \(-1\) is 'cancelled out', and \(r\) stands alone on one side of the equation. The equation's balance is maintained because whatever operation is performed on one side is also done on the other side.

Simplifying the equation is a technique to rewrite the equation in the simplest form possible. This process usually involves combining like terms and performing basic arithmetic to decrease complexity and clarify the solution.

In our equation, after adding \(1\) to both sides, we must combine like terms. By simplifying \(r-1+1\), the ones cancel each other out, leaving just \(r\) on the left side. On the right side, \(16+1\) simplifies to \(17\). The process of simplification leads to the final solution, where the equation \(r=17\) clearly displays the variable's value without any other numbers or operations, making it apparent that \(r\) equals \(17\).