91Ó°ÊÓ

Equations are mathematical statements that assert the equality of two expressions. They are comprised of variables, numbers, and operational symbols such as plus, minus, or even exponentiation.
In the exercise, the equation we start with is:

• \(x + y^{3} = 8\)

This equation contains two variables: \(x\) and \(y\). By plugging different numbers for \(x\) or \(y\), you can find corresponding values for the other variable, maintaining the equality intact.
In basic algebra, equations are essential as they allow us to solve for unknown variables. The goal often is to express one variable in terms of the other, or to determine the specific values of these variables that satisfy the equation.

Solving for a variable such as \(y\) means isolating it on one side of the equation, typically the left side, to express it in terms of other variables or constants on the right side. This process helps us understand how \(y\) behaves when influenced by changes in the other variable, in this case, \(x\).
In our example, the first step to solve for \(y\) in \(x + y^3 = 8\) is rearranging the equation by subtracting \(x\) from both sides:

• \(y^3 = 8 - x\)

This step allows us to isolate \(y^3\) on one side of the equation.
Next, you solve for \(y\) by taking the cube root of both sides. This involves applying the inverse operation of cubing, confirming that \(y\) can be expressed as a function of \(x\):

• \(y = \sqrt[3]{8 - x}\)

By isolating \(y\), it becomes clear that each \(x\) value will determine a unique \(y\) value, validating \(y\) as a function of \(x\).

The cube root of a number is an interesting concept, especially when dealing with negative results, as both positive and negative numbers can have cube roots.

• The cube root of a number \(n\) is a value that, when multiplied by itself twice (\(n \times n \times n\)), gives the original number \(n\).

Unlike square roots, cube roots are unique for every number, making them particularly useful in functions like ours.
Consider the expression \(y = \sqrt[3]{8 - x}\), which represents the cube root of the expression \(8 - x\). Whether \(8 - x\) is positive or negative, there will always be a unique cube root value of \(y\) corresponding to each value of \(x\).
This uniqueness is what confirms that \(y\), as expressed, is indeed a function of \(x\). Cube roots help us understand complex relationships in a more linear and predictable manner, often simplifying otherwise difficult equations.