91Ó°ÊÓ

Algebra is like the language of mathematics. It involves using letters and symbols to represent numbers and express equations. In the context of the exercise, we are dealing with the linear equation defined by the function \( f(x) = 5x - 2 \). This equation is in the form of \( f(x) = ax + b \), which is a standard form for linear functions. Here:

• \( a = 5 \)
• \( b = -2 \)

These constants \( a \) and \( b \) define the slope and y-intercept of the line that's represented by the equation.
The goal is to find the value of \( x \) that makes \( f(x) = 3 \). In algebra, this means manipulating the equation to isolate \( x \) and find its value. Understanding these basic operations is key to algebra because they allow us to rearrange equations and solve for unknown values.

Function evaluation is the process of finding the output of a function for a specific input. When we have a function like \( f(x) = 5x - 2 \), evaluating it for a particular \( x \) means substituting \( x \) into the equation and calculating the result.
In this exercise, we set \( f(x) = 3 \) and then determine what \( x \) must be so that when substituted into \( f(x) \), the output equals \( 3 \). This is the reverse process of what you usually do in function evaluation, as instead of substituting a given \( x \) and finding \( f(x) \), you have \( f(x) \) and need to find \( x \).
Function evaluation is central to understanding how the inputs and outputs of functions relate to each other, which is crucial in not just algebra but in all calculus and advanced mathematics as well.

Solving equations involves finding the values of the variables that make the equation true. This task is often the heart of many algebra problems.
In the given exercise, we started with the equation \( 3 = 5x - 2 \). The goal is to solve for \( x \). Here are the steps we took:

• Add \( 2 \) to both sides to cancel out \( -2 \) on the right: \( 3 + 2 = 5x \).
• Simplify to \( 5 = 5x \).
• Divide both sides by \( 5 \) to isolate \( x \): \( \frac{5}{5} = x \).

Through these steps, we discover that \( x = 1 \).
Solving linear equations is straightforward once you understand the underlying principles. Always aim to isolate the variable by performing inverse operations on both sides of the equation. This skill is foundational and applies to more complex equations and systems, forming a backbone of mathematical problem-solving.