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Solving for a variable in an equation is like finding a missing piece of a puzzle. Our goal is to find the value of the unknown that makes the equation true. When solving equations, each move should help us get one step closer to seeing that missing piece clearly. In the given exercise, the equation is originally presented as \(10x - y = 2\), making it a little tricky because it's not in the simplest form to find \(y\) immediately.

When we solve for a variable, such as \(y\), we perform operations that "undo" what’s around \(y\). Each action we take should align with keeping the equation balanced. Imagine a see-saw; if you change something on one side, you must do the same on the other side to keep it even.

Here, solving for \(y\) meant rearranging the equation until \(y\) was all by itself on one side of the equation, revealing its relationship with the rest of the equation.

Equation manipulation is the art of transforming an equation into simpler or more useful forms. It involves performing operations like addition, subtraction, multiplication, or division to reshape how equations appear. The main rule is always to do the same thing to both sides of the equation.

In our specific example, manipulation was key. The original equation, \(10x - y = 2\), isn't in a form that easily tells us what \(y\) is. To manipulate this equation, we subtract \(10x\) from both sides, shifting \(10x\) and isolating \(-y\) on one side.

Steps involved in manipulating equations often include:

• Adding or subtracting terms from both sides
• Multiplying or dividing both sides by the same number
• Rearranging terms to make the equation look cleaner or simpler

By understanding these steps, we gain control over the form and structure of any equation we encounter.

Isolation of variables is about getting the unknown by itself on one side of the equation. Think of it as making sure our variable stands alone, all neat and independent so that we can easily "read" its value or its role in the equation.

In the given exercise, once we had \(-y = -10x + 2\), \(y\) wasn't quite alone yet because it had a negative sign attached to it. The final step in variable isolation was flipping the sign. To make \(y\) positive, both sides of the equation were multiplied by \(-1\). This produced the clean form \(y = 10x - 2\).

The aim of isolation is to make equations easier to solve by simplifying how the variables and numbers relate to each other. It’s a critical skill in algebra, helping to peel away layers of complexity. This technique is like adjusting the lens on a camera, allowing us to focus on what's important: the isolated variable.