91Ó°ÊÓ

Solving equations is all about finding the value of the variable that makes the equation true. In the exercise above, we want to solve the linear equation \(6y - 1 = 16\). This means we need to determine what number \(y\) can be replaced with, so both sides of the equation are equal. Understanding the process of solving equations is like following the instructions in a recipe. Each step brings us closer to the final answer, just as each ingredient adds to the overall dish. For this equation, following the correct steps systematically allows us to reach \(y = \frac{17}{6}\). It’s important to remember that equations maintain balance. Whatever operation is applied to one side must also be applied to the other. This keeps the equation valid and helps us find the solution step by step.

Variable isolation is the process of rearranging the equation to get the variable by itself on one side. This is crucial because it helps us clearly see the value of the variable.To isolate \(y\) in the equation \(6y - 1 = 16\), we began by moving the constant \,\(-1\), to the other side.

• Addition or subtraction is often used to cancel out constants, such as \,\(-1\), thereby isolating terms with the variable.
• The step \(6y - 1 + 1 = 16 + 1\) simplifies to \(6y = 17\), showing how we bring the variable terms to one side.

By the end of isolation, the variable is ready to be solved, bringing us a step closer to determining its value. This skill is fundamental in algebra, as it forms the basis of transforming and solving any equation.

Algebraic manipulation involves using operations to rearrange terms and solve equations. It includes techniques like adding, subtracting, multiplying, and dividing both sides of an equation. In our problem, we used algebraic manipulation to change the equation from \(6y = 17\) into \(y = \frac{17}{6}\) through division.

• The act of dividing both sides by \(6\) ensures we directly solve for \(y\), giving us the exact value needed.
• All actions must keep the equation balanced, which is why every operation was conducted on both sides.

Mastering algebraic manipulation is key to simplifying and solving more complex problems later on. By systematically applying these techniques, even the toughest equations become manageable.