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When solving equations, the goal is to find the value of the unknown variable that makes the equation true. This generally involves two key actions: performing operations that simplify the equation and maintaining balance by performing the same operations on both sides. For example, in the equation \(-55 = 3w + 5\), we need to determine what value of \(w\) satisfies the equality.

To solve an equation, you often begin by simplifying it as much as possible. Simplification involves combining like terms, distributing where necessary, and eliminating any unnecessary numbers or variables. By doing these steps, you walk progressively closer to isolating the variable. Solving equations is a fundamental skill in algebra, and mastering it is essential for tackling more complex mathematical problems.

Isolation of the variable is a critical step in solving equations. This process involves rearranging the equation until the variable is by itself on one side of the equation. For example, in the equation \(-55 = 3w + 5\), the goal is to get \(w\) by itself on one side, preferably the left side.

Here's how you can achieve this:

• Identify the term containing the variable. In our example, it is \(3w\).
• Move any other terms to the opposite side by performing inverse operations. Since the term with \(w\) is accompanied by a \(+5\), you would subtract 5 from both sides to eliminate it, resulting in \(-60 = 3w\).
• Continue to simplify until the variable stands alone. With \(3w\) on one side, divide each side by the coefficient of \(w\) to isolate it completely, thereby solving \(3w = -60\) by dividing through by 3 to obtain \(w = -20\).

Algebraic manipulation refers to the process of rearranging and simplifying algebraic equations through various operations to solve them or to put them in a more useful form. In the context of linear equations, manipulation often involves addition, subtraction, multiplication, and division.

In the exercise \(-55 = 3w + 5\), we used algebraic manipulation primarily through the following steps:

• **Addition/Subtraction:** These operations were used to move terms from one side of the equation to the other, such as subtracting 5 from both sides to eliminate the constant term.
• **Division:** This was used to simplify the equation by eliminating the coefficient of the variable. Dividing both sides by 3 was the key to isolating \(w\) on its own.

Algebraic manipulation is a powerful tool that allows for a greater understanding of equations, enabling you to break down complex problems into simpler, solvable tasks. Mastery of these techniques results in more efficient problem-solving, especially critical in algebra.