91Ó°ÊÓ

A two-column proof in algebra is a structured way of solving equations, where the left column lists each step of the solution and the right column explains the reason or property used for that step. This method is very helpful in organizing your thoughts and ensuring that each operation is valid. In this exercise, the two-column proof sets the framework by clearly defining the logical steps needed to solve \(-4x + 5 = -15\). A key advantage is that it prevents mistakes, as each change to the equation is justified. This systematic approach also builds a strong understanding of how to manipulate equations, so you fully comprehend why each step is necessary. For example, when isolating the variable \(x\), you ensure each action is matched with an explanation, such as applying the Addition or Division Property of Equality.

The Addition Property of Equality provides the foundation for solving many algebraic equations. This property states that if you add (or subtract) the same number to both sides of an equation, the sides remain equal. It sounds simple, but it's crucial: equations stay balanced like a scale. In solving \(-4x + 5 = -15\), this property helps us eliminate the constant \(+5\) on the left side. By subtracting \(5\) from both sides of the equation, we simplify to \(-4x = -20\). This step makes the equation easier to solve because it removes the integers not associated with \(x\), letting you focus on the term you need to work with.

The Division Property of Equality allows you to divide both sides of an equation by the same non-zero number without altering the equation's balance. This property is particularly useful for isolating a variable, which is often the goal when solving equations in algebra. In the exercise example, following the application of the Addition Property of Equality, we are ready to apply the Division Property of Equality to solve for \(x\). Here, you'll divide both sides of \(-4x = -20\) by \(-4\). This action gives \(x = \frac{-20}{-4}\), which simplifies to \(x = 5\). Using this property ensures every division is valid and maintains the equation's integrity, allowing for precise and confident solving.