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An equation is a mathematical statement that asserts the equality of two expressions. In simpler terms, an equation is like a balance; what is on one side of the equal sign is equivalent to what's on the other. For example, in the equation \( x^{2} + y = 4 \), the expression \( x^{2} + y \) equals 4. This means that any solution we find will keep both sides of the equation balanced and true.
Understanding how to manipulate equations is crucial because they are the building blocks of functions, inequalities, and more complex algebraic structures. Here are some tips to remember when working with equations:

• Isolate the variable you need to solve for by using inverse operations like addition or subtraction.
• Keep the equation balanced by doing the same operation on both sides.

In our original problem, we started with the equation \( x^{2} + y = 4 \) and worked towards isolating \( y \) to understand its relationship with \( x \).

To express \( y \) in terms of \( x \), you need to rewrite the equation, isolating \( y \) on one side. This process is often referred to as 'solving for y'. In our example of \( x^2 + y = 4 \), solving for \( y \) involves a few simple steps:

• First, identify what needs to be moved to the other side of the equation. Here, we have \( x^2 \) next to \( y \).
• Subtract \( x^2 \) from both sides to clear \( x^2 \) from \( y \). This keeps the equation balanced.
• Your new expression is now \( y = 4 - x^2 \).

This formula, \( y = 4 - x^2 \), gives us a clear picture of how \( y \) changes as \( x \) changes, making it easier to identify the properties of the potential function.

The function test helps to determine if an equation represents a function. A basic principle of functions is that for each input \( x \), there should be exactly one output \( y \). In simple terms, no single \( x \) value should lead to two or more different \( y \) values.
For our transformed equation \( y = 4 - x^2 \), we apply this test by checking the output for various inputs:

• Try different values of \( x \) (e.g., \( x = 1, -1, 0 \)) and calculate \( y \) using our equation.
• Ensure each \( x \) results in exactly one \( y \) value.

Since for every unique \( x \) we find one unique \( y \), \( y \) is indeed a function of \( x \). This understanding is important because it defines how dependent and independent variables interact in equations and real-world applications.