91Ó°ÊÓ

In solving algebraic equations, one of the primary goals is to isolate the variable you’re solving for. In other words, you need to get the variable by itself on one side of the equation. This is crucial because it allows you to easily determine its value.
For example, in the given exercise, we aim to isolate \(y^2\). We start with the equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). The first step is to remove the term \(\frac{x^2}{a^2}\) from the left side. We do this by subtracting it from both sides, transforming the equation into: \(\frac{y^2}{b^2} = 1 - \frac{x^2}{a^2}\). By isolating \(y^2\) using subtraction, we've made the equation simpler to work with in subsequent steps.

Algebraic manipulation involves using basic algebraic properties to simplify an equation or expression. This often involves operations like addition, subtraction, multiplication, and division.
In our exercise, once we have isolated \(y^2\) as \(\frac{y^2}{b^2} = 1 - \frac{x^2}{a^2}\), we need to clear the fraction involving \(y^2\).
The simplest way to do this is by multiplying every term in the equation by \(b^2\) to eliminate the fraction: \(y^2 = b^2 (1 - \frac{x^2}{a^2})\). This step ensures that \(y^2\) stands alone on one side of the equation, making the equation ready for further manipulation like finding roots. Algebraic manipulation is an essential skill as it lays the foundation for solving complex equations.

The square root property allows us to solve equations involving squared variables by taking the square root of both sides. Remember, whenever you take the square root, you must consider both the positive and negative possibilities.
In the exercise, after manipulating the equation to \(y^2 = b^2 (1 - \frac{x^2}{a^2})\), we apply the square root property.
Taking the square root of both sides, we find: \(y = \pm \sqrt{b^2 (1 - \frac{x^2}{a^2})}\). This means "\(y\)" can be either the positive or negative root of the resulting expression.
This step is critical in algebra as it finalizes the solution, giving you the actual value or values of the variable in question. Always remember to check both the positive and negative outcomes when using the square root property.