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In solving equations, the first major step often involves the isolation of variables. The goal here is to get the variable by itself on one side of the equation. This allows us to find its value. To achieve this, we perform operations that reverse or undo those done to the variable. These operations include addition, subtraction, multiplication, and division.

In our exercise, we start with the equation:

• \(6x - 3 = 63\)

To isolate the term \(6x\), we add 3 to both sides. This operation helps remove the constant from the variable term. The equation becomes:

• \(6x = 66\)

By isolating \(x\), we effectively simplify the task, setting us up to solve for \(x\) itself. The actions we take must maintain the balance of the equation. Remember, whatever you do to one side must be done to the other. This keeps the equation true.

After finding a potential solution, it's important to perform arithmetic checks. This ensures that our solution indeed satisfies the original equation. Mistakes in calculations can happen, and checking acts as a safeguard.

For instance, with \(x = 11\), we substitute this value back into the original equation:

• \(6(11) - 3\)

The arithmetic here involves simple operations:

• Calculate \(6 \times 11 = 66\)
• Then subtract 3 from 66 to get \(63\)

Finally, we verify that both sides of the equation are equal, confirming that \(x = 11\) is correct. Through these checks, we ensure accuracy and reliability in our solution process. Always take this step seriously, as it determines correctness and bolsters confidence in your answer.

Linear equations are equations of the first degree, meaning they involve variables raised to the power of one. The standard form of a linear equation is \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

The solution to a linear equation aims to find the value of the variable \(x\) that makes the equation true. Our exercise is a classic case of a linear equation:

• \(6x - 3 = 63\)

Simplifying and solving, as outlined in the steps, shows the core idea behind dealing with linear equations.

• They are straightforward because they form a line when graphed.
• The steps we perform in solving them maintain balance and equality.

Understanding linear equations is fundamental since they are foundational in algebra. Solving them involves systematic steps that apply to more complex equations later in mathematics.