91Ó°ÊÓ

Isolating a variable is a fundamental technique in algebra that involves rearranging an equation so that a single variable stands alone on one side of the equal sign, making it the subject of the equation. The purpose of this method is to make it easier to understand the relationship between variables, often to solve for a particular one.

In the context of our problem, we aim to solve for the variable \(y\). This means we want \(y\) by itself on one side of the equation and all other terms on the opposite side. To achieve this, we use algebraic operations that maintain the balance of the equation -- whatever we do to one side, we must also do to the other.

Starting with the rearranged standard form equation \(x - y + 4 = 0\), we subtract \(x\) and add \(4\) to both sides to isolate \(y\), ultimately arriving at \(y = x - 4\). This simplifies the original equation and clearly shows how \(y\) is defined in terms of \(x\).

The standard form of an equation is a way of writing it so that all terms are aligned on one side of the equal sign, usually with the highest degree term first, followed by lower degree terms in descending order. Standard form is often preferred because it provides consistency and makes it easier to compare and manipulate equations.

For linear equations, like the one in our exercise, the standard form is typically \(Ax + By + C = 0\), where \(A\), \(B\), and \(C\) are real numbers, and \(A\) or \(B\) (but not both) could be zero. This form clearly separates variables from constants and makes it straightforward to solve for any variable.

In our case, the equation was initially given as \(4 - y + x = 0\). Following standard form conventions, we re-order the terms to get \(x - y + 4 = 0\). This step sets the stage for further manipulation designed to solve for the desired variable.

Algebraic manipulation encompasses the various techniques used to transform and simplify equations. These methods include adding, subtracting, multiplying, or dividing both sides of the equation by the same non-zero number, as well as using the distributive property and combining like terms.

When solving for a variable such as \(y\), you'll often employ these techniques to move terms involving other variables and constants to the opposite side of the equation. It's important to perform the same operation to both sides to keep the equation balanced. This process ensures that the equivalence set by the initial equation remains true throughout our manipulations.

For example, to solve for \(y\) in the standard form equation \(x - y + 4 = 0\), we add \(y\) to both sides to get \(x + 4 = y\), which can also be written as \(y = x - 4\) after subtracting \(4\) from both sides. The solution clearly defines \(y\) in terms of \(x\), illustrating a direct application of algebraic manipulation techniques to isolate the desired variable.