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When you encounter a linear equation, the main goal is often to solve for a particular variable. In this case, we want to find the value of \(y\) in terms of \(x\). To achieve this, we need to perform operations that simplify the equation in such a way that \(y\) is isolated on one side. Breaking down the process into smaller steps helps make this more manageable. For example, our equation starts as \(2x - 5y = 10\). The end goal is to solve this equation for \(y\), providing an expression that tells us exactly how \(y\) changes according to \(x\). This process involves skills like understanding which operations to apply to both sides and knowing how algebraic manipulations maintain the equation's balance.

To isolate \(y\), we need to rearrange the equation so that \(y\) appears by itself on one side of the equal sign. You start this by moving all other terms to the opposite side. For the equation \(2x - 5y = 10\), we need to move \(2x\) to the other side, resulting in \(-5y = -2x + 10\). This re-positioning helps us focus only on the term with \(y\). The operation we use here is subtraction, as it effectively 'cancels out' \(2x\) from the side with \(y\). The key is ensuring every operation keeps the equation balanced, applying the same change to both sides whenever altering it.

Equation rearrangement involves structuring an equation so that it's ready for the final solution steps. Lines of mathematical logic guide this process.In our example, we took the initial equation \(2x - 5y = 10\) and rearranged it to \(-5y = -2x + 10\). Here, the decision to shift terms assists in setting the stage for isolating \(y\). Rearranging an equation:

• Focuses the work on a single variable.
• Utilizes arithmetic operations like addition or subtraction.
• Ensures an equation remains balanced and solvable.

This step is crucial as it simplifies subsequent operations needed for solving for the variable.

After moving all other terms away from \(y\), you are left with a coefficient—here, \(-5\)—that needs dealing with. Coefficient manipulation is about managing this number. For our equation \(-5y = -2x + 10\), divide every term by \(-5\) to isolate \(y\). The operation changes the equation to \(y = \frac{2x}{5} - 2\). Key points in manipulating the coefficient:

• Divide every term by the coefficient to isolate the variable.
• Ensure the division correctly affects all parts of the equation.
• Remember that dividing by a negative flips the signs in the equation.

This manipulation step ensures \(y\) stands alone, giving us a clear expression of \(y\) based solely on \(x\).