Solve for f:
$f - 6 = -2f + 3(f - 2)$
The given equation is \(f - 6 = -2f + 3(f - 2)\). Our goal is to solve for the variable \(f\) and determine the nature of its solution set. This is a linear equation where we need to simplify both sides to see how the variable \(f\) behaves.
First, we focus on simplifying the right side of the equation. We distribute the constant \(3\) across the terms inside the parentheses:
\[3(f - 2) = 3f - 6\]
Substituting this back into the original equation gives:
\[f - 6 = -2f + 3f - 6\]
Now, we combine the like terms on the right side. Specifically, we have \(-2f + 3f\), which simplifies to \(f\). Thus, the equation becomes:
\[f - 6 = f - 6\]
This result shows that the expression on the left side of the equation is identical to the expression on the right side.
Since the equation simplifies to \(f - 6 = f - 6\), it is true for any real number \(f\). This type of equation is called an identity. Because the equality holds regardless of the value assigned to \(f\), there are infinitely many solutions. We can express this conclusion by stating that the solution set is all real numbers, denoted as \(f \in \mathbb{R}\).
Understanding identities is crucial in fields like engineering and economics, where you might encounter models that simplify to statements like 'total cost = total cost' regardless of production volume. For instance, if a business owner calculates their profit using two different methods that are algebraically equivalent, discovering an identity ensures that both methods will yield the same result under any circumstances, providing reliability in financial analysis and strategic planning.