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Solving for a variable means finding the value of that variable in terms of the other quantities given in the equation. In the exercise we are discussing, our objective is to express the variable \( y \) using the variable \( x \).

When solving for a variable, like \( y \) in the equation \( x = -6y \), the first step is to rearrange the equation so that \( y \) is by itself on one side. This crucial step allows us to understand how \( y \) relates to the other components in the equation. Each operation you perform on one side of the equation must be performed on the opposite side to maintain equality.

Once \( y \) is isolated, the equation is considered solved for \( y \), offering a clear insight into the relationship between \( x \) and \( y \). This clarity is essential in understanding the broader context in applied problems, making the information actionable.

Isolating a variable involves rearranging an equation so that the variable you are solving for is alone on one side of the equation. This usually involves performing operations such as addition, subtraction, multiplication, or division.

In our example, to isolate \( y \) in the equation \( x = -6y \), we divide both sides by \(-6\). We carry out this division because \( y \) is currently being multiplied by \(-6\), and dividing both sides by \(-6\) cancels out this multiplication:

\[ y = \frac{x}{-6} \]

Isolating the variable is a principle of algebra used frequently in solving all types of equations. It gives you a direct answer related to the variable of interest, making it imperative for problem-solving and mathematical reasoning.

Simplifying expressions is about transforming an equation or an expression into its simplest form, making it easier to understand or use.

The expression \( y = \frac{x}{-6} \) can be simplified. We can see that dividing by a negative number can be presented more clearly by rewriting it:

\[ y = -\frac{x}{6} \]

This form is considered simplified because it eliminates any unnecessary fractions or negative signs that could complicate interpreting the expression.

Simplifying also often makes further calculations more straightforward and reduces the chances of errors. Being skilled at simplifying expressions is valuable as it enhances understanding and clarity, making the math less intimidating and more approachable for anyone.