Understanding Identities and Equivalencies in Algebra

When discussing algebraic equations, it is crucial to distinguish between identities and equivalencies. An identity is an equation that is true for all possible values of the variables involved, whereas an equivalency is an equation that is true when simplified or transformed through algebraic operations.

What is an Identity?

Identities are equations that are true for every possible value of the variables. For instance, the identity

sin2x (cos^2x) 1

holds true for any value of (x). This means that no matter what value we assign to (x), the left-hand side will always equal the right-hand side. Another example of an identity is

sec2x tan2x 1

and

cos2x sin2x 1

These identities are fundamental properties of trigonometric functions.

Are y 2x3 and 2y 4x? Considered Identities?

Let's examine the given equations y 2x3 and 2y 4x?. These equations are not equivalent or identical equations. To see why, let's simplify the second equation:

Starting with 2y 4x?, we can divide both sides by 2:

y 2x?

This shows that y 2x3 and y 2x? are different equations, as they do not simplify to the same form. Therefore, although they can represent the same line for specific values of (x), they are not identical equations. An identity would hold true for all (x), and these equations do not satisfy that condition.

Simultaneous Equations and Infinitely Many Solutions

Now, consider a system of equations:

y 2x3

2y 4x?

When we solve the second equation for (y), we get:

y 2x?

This means the system reduces to:

2x3 2x?

or simplifying further:

x3 x?

This can be rewritten as:

x3(1 - x3) 0

Thus, the solutions are (x 0) and (x 1). For each of these values, we can substitute back to find corresponding (y)-values:

This results in the points ((0, 0)) and ((1, 2)), meaning there are only two solutions, not infinitely many. Therefore, this system of equations is not equivalent to the identity equation.

Are y 2x3 and 2y 4x? Equivalent?

An equivalency occurs when two equations represent the same line, as they do when simplified through algebraic operations. For example, if we have:

y 2x3

and

2y 4x? (which simplifies to y 2x?)

These equations are different, as they represent different lines unless (x 0). In this case, they would indeed be equivalent at the point where (x 0), but they are not equivalent as general equations.

To summarize, the key distinction is that an identity must hold true for all values of the variables involved, while an equivalency is a relationship that holds under certain conditions.