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algebra is used to isolate and find the value of a variable from an equation like the one given. A fundamental step in solving algebraic equations is keeping both sides of the equation balanced. For example, when you move a term from one side of the equation to the other, you must perform the opposite operation. This ensures the equation remains true.

Dealing with fractions is a common task in algebra. You need to know how to add, subtract, multiply, and divide fractions to solve equations correctly. In the given exercise, fractions are manipulated several times. Finding a common denominator is crucial for adding or subtracting fractions. This involves identifying the least common multiple (LCM) of the denominators. For instance, while subtracting \(\frac{4}{3} - \frac{1}{2} \), the common denominator is 6, which transforms the fractions into \(\frac{8}{6} - \frac{3}{6} = \frac{5}{6} \). Finally, simplifying fractions to their lowest terms makes your solution more readable and accurate. Simplifying \(\frac{-15}{6} \) to \(\frac{-5}{2} \) is a crucial step.

Solving linear equations systematically is essential for clarity and correctness. Here is a streamlined process:

• Identify the equation.
• Transpose terms to isolate the variable term on one side. This usually involves moving constants or variables across the equal sign.
• Simplify the resulting expression. This could mean finding a common denominator as seen in subtracting fractions.
• Perform operations to isolate the variable. This might involve multiplication or division, depending on what is necessary to solve for the variable.
• Simplify your final answer, including reducing fractions to their simplest form when applicable.

These steps help break down complex problems into manageable tasks, easing the process of finding the solution.