91Ó°ÊÓ

When solving a linear equation like the one given, the goal is to find the value of a specific variable. In this case, we want to solve for the variable \( y \). This means that we need to rewrite the equation in terms of \( y \) so that we can determine its value based on the information we have. To accomplish this, we often perform various algebraic operations to manipulate the equation and "free" \( y \) from any other terms that might be influencing it. By strategically moving terms from one side of the equation to the other, we can simplify our task of solving for \( y \). Throughout this process, keep in mind our main objective: rewrite the equation so that \( y \) stands alone on one side, showing its value in terms of other known values and variables.

Isolation of terms is a key concept in solving equations. It involves getting the variable you are solving for by itself on one side of the equation. In the given equation \( x = -2y - 7 \), our task is to move all other terms away from \( y \). This process typically involves performing the opposite of whichever operations are being applied to \( y \).Let's break it down:

• Since \( y \) is being multiplied by \(-2\), we aim to eventually cancel this operation by dividing by \(-2\).
• First, remove the constant term \(-7\) by adding \(7\) to both sides of the equation, resulting in \( x + 7 = -2y \).

These steps allow us to systematically isolate \( y \) from the other terms so that we can focus entirely on \( y \)'s value. Always ensure that any operations applied to one side of the equation are equally applied to the other side to maintain balance.

Algebraic manipulation involves using basic algebraic rules to alter the form of an equation without changing its intrinsic properties. These manipulations allow us to solve for a variable effectively.In our exercise, once we isolated \( y \), we had an equation \( x + 7 = -2y \). The presence of \(-2\) multiplying \( y \) required us to divide both sides by \(-2\) to fully isolate \( y \). Let's go through this:

• Division is applied here because it counters multiplication. By dividing both sides of the equation by \(-2\), we solve for \( y \).
• In our specific equation, the division changes it to \( y = -\frac{x + 7}{2} \).

Algebraic manipulation hinges on maintaining balance. Whatever you do to one side of the equation, you must equally do to the other. This principle ensures that even after manipulation, the equation's integrity remains intact, leading to a correct solution.