91Ó°ÊÓ

When we talk about linear equations, we refer to algebraic statements that involve constants and variables, specifically with each variable to the first power. A linear equation looks something like this: \(ax + b = 0\), where \(a\) and \(b\) are constants, and \(x\) is the variable we aim to find. In the exercise \(5x - 3 = 4x\), we have a clear example—a simple linear equation with one variable.

Linear equations are the foundation of algebra and are used to represent straight lines on a graph. They have at most one solution, like the one we find in the provided example. The power of linear equations lies in their simplicity and their wide range of applications in fields such as physics, engineering, economics, and beyond.

Delving into algebraic equations reveals a broader scope than linear equations alone. These can come in many different forms, including quadratic, cubic, and higher-degree equations. An algebraic equation is essentially a mathematical statement that uses numbers, variables, and operators (like +, -, *, and /) to convey a relationship where two expressions equal each other.

They are pivotal in solving problems that describe real-world scenarios, like calculating rates, areas, or even population growth. Our example, \(5x - 3 = 4x\), is a linear type of algebraic equation, which is vital in developing problem-solving skills in algebra.

The procedure of solving an equation step by step is a fundamental skill in algebra. The process aims to isolate the variable on one side of the equation to find its value. For the equation \(5x - 3 = 4x\), we have a concise series of steps:

• Isolate the Variable: Subtract \(4x\) from both sides to get \(x - 3 = 0\).
• Solve for the Variable: Add 3 to both sides, which results in \(x = 3\).

This methodical approach ensures that we maintain the balance of the equation, meaning whatever operation we do to one side, we must do to the other. This principle keeps the equation 'true' and leads us to the correct solution.

Once a solution is found, checking it is the final step to guarantee accuracy. This involves substituting the solution back into the original equation to ensure the left and right sides are equal. In the exercise we're examining, after finding the solution \(x = 3\), we check it by replacing \(x\) with 3 in the original equation: \(5 * 3 - 3\) should equal \(4 * 3\), which simplifies to \(12 = 12\).

This step confirms \(x = 3\) as the correct solution. The checking process is essential since it helps catch any mistakes made during the solving process and validates our work, boosting our confidence in the solutions we present.