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Equations featuring variables are at the core of algebra and are a fundamental part of solving mathematical problems. Variables are symbols, usually represented by letters like \( x \), \( y \), or \( z \), that stand in for numbers we want to find. These letters make it possible to create mathematical statements representing dynamic relationships.
For example, in the equation \( 6x - 4 = 2x \), the variable \( x \) represents an unknown number. The task is to find a value for \( x \) that makes both sides of the equation equal.

• The equation can have one or more variables
• The goal is to find the numerical value of these variables
• The result should satisfy the original equation

Understanding equations with variables is essential because they appear in many types of mathematical problems and real-world situations. You will often rearrange and manipulate these equations to find the solution.

Solving equations requires a step-by-step approach to find the solution effectively. Breaking down the process ensures clarity and helps avoid mistakes along the way. Let's review how a step-by-step solution works using the equation \( 6x - 4 = 2x \).

• Identify the equation parts: Understand what terms involve variables and which are constants. This equation has variable terms \( 6x \) and \( 2x \), and a constant \(-4\).
• Move all variable terms to one side: Subtract \( 2x \) from both sides to move all \( x \) terms to the left side, resulting in \( 4x - 4 = 0 \).
• Isolate the variable term: Adjust the equation by adding or subtracting constants on both sides. Adding 4 to both sides gives \( 4x = 4 \).
• Solve for the variable: Divide each side by the coefficient of the variable, \( 4 \), giving \( x = 1 \).
• Verify the solution: Substitute \( x = 1 \) into the original equation. Check if it makes both sides equal, which confirms the solution is correct.

These steps not only help understand the current problem but also train you in solving other similar problems by following a systematic approach.

Isolating variables is a crucial technique that makes solving equations straightforward. It involves arranging the equation in such a way that the variable you are solving for stands alone on one side of the equation. Let's look at how this works in the equation \( 6x - 4 = 2x \).
To isolate \( x \):

• Start by removing any terms on the same side as \( x \) that are not with \( x \). In our equation, subtract \( 2x \) from both sides so all terms involving \( x \) are on one side, simplifying to \( 4x - 4 = 0 \).
• Next, deal with the constant term by adding \( 4 \) to both sides, resulting in \( 4x = 4 \).
• Finally, divide each side by the coefficient of \( x \), which here is \( 4 \), resulting in \( x = 1 \).

By ensuring the variable is isolated, the problem becomes much simpler, allowing us to clearly see the solution. Mastering this concept is essential for solving more complicated equations that involve multiple steps and operations.