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Solving equations is a fundamental part of algebra. The goal is to find the value of the variable that makes the equation true. Let's consider the equation \(-5x + 20 = -50\). When solving any equation, the first step involves clearing any constants that are tied up with the variable. This is often referred to as 'isolating the variable.' By following systematic steps, you can manage to keep the equation balanced.
To solve, you aim to have the variable mentioned on one side and constants on the other. This simplifies identifying the solution. Maintaining a balanced equation is key, meaning if an operation is performed on one side, it must also be done on the opposite side. Only through this consistent balancing act can the correct solution be arrived at. Solving equations efficiently requires understanding both the equation and the properties of operations.

Algebraic manipulation plays a crucial role in solving equations. In \(-5x + 20 = -50\), it involves strategic actions like adding, subtracting, multiplying, or dividing both sides by the same number to maintain the equation's balance.

• Addition/Subtraction: In this problem, we begin by removing the constant \(+20\) from the left by subtracting it. This simplifies our equation to \(-5x = -70\).
• Division/Multiplication: Next, to isolate \(x\), divide each side by \(-5\). This effectively cancels out the coefficient of \(x\), yielding \(x = 14\).

Algebraic manipulation involves a series of steps or operations that change the structure of an equation while keeping both sides equivalent. Mastering these manipulative techniques allows you to transform and simplify equations, making it easier to solve for the unknown.

To effectively solve equations, it is essential to have a good grasp of mathematical operations. These operations form the backbone of the methods used to isolate variables in an equation.
In our example, you executed two operations:

• Subtraction: This is used to eliminate constants from the left side, moving it to the right to isolate the term with \(x\).
• Division: Following subtraction, division by \(-5\) reduces the expression to a simple \(x = 14\), finding us the value of \(x\).

Understanding when and how to employ each operation allows any algebraic equation to become solvable. Each operation works within a set of rules, ensuring you're steering the equation closer to its simplest form where solving becomes straightforward. This understanding is what transforms theoretical mathematics into practical problem-solving abilities.