91Ó°ÊÓ

When solving algebraic equations, one key step can often be isolating the variable we are interested in.
In this problem, we want to see if the equation defines y as a function of x.
To begin, we start with the equation: \[ x^2 - 4y^2 = 1 \] Our goal is to isolate y.
We rearrange the equation to isolate the term containing y: \[ x^2 - 1 = 4y^2 \] This step was necessary to bring all the non- y terms to one side of the equation.
Afterwards, we divide by 4 to further isolate y^2: \[ y^2 = \frac{x^2-1}{4} \] Finally, we take the square root of both sides to solve for y: \[ y = \frac{eg egegeg egegegegeg \sqrt{x^2-1}}{2} \]
Important: When you isolate a variable, remember to perform the same operation on both sides of the equation.
This gets us closer to knowing if we have a function.

Solving equations involves undoing operations to isolate the variable.
In our case, solving for y meant isolating y on one side of the equation. Let' s take a closer look: \[ x^2 - 4y^2 = 1 \ to \ x^2 - 1 = 4y^2 \]
We subtracted from both sides to move non-y terms left.
Next, divide by 4: \[ y^2 = \frac{x^2 - 1}{4} \]
Taking the square root simplifies the equation:
y is \[ \frac{eg egegeg\sqrt{x^2-1}}{2} \]

A critical part is understanding the function criterion.
A function means each input (x) maps to one output (y).
In our example, \[y = \frac{- eg(x^2-1)}{2} \]
provides two outputs for a single x.