91Ó°ÊÓ

Simplification is the first step in solving any equation. It involves combining like terms and reducing expressions to their simplest form. In this exercise, the equation provided is \(2z = 12 - 14\).

We look at the right side of the equation, which has the expression \(12 - 14\). By performing the subtraction, we get a simplified expression of \(-2\). Now the equation is much easier to handle: \(2z = -2\). By simplifying, we've transformed a potentially confusing equation into one that is straightforward and poised for the next steps. Simplification clears up any clutter and allows you to see what you're really working with. Keep in mind that every simplification step is like making your path clearer on your journey to finding the solution.

Isolation of a variable is a critical step when solving equations, as the goal is to have the variable by itself. When we isolate a variable, we get a clearer view of what the variable equals.

In our example, the equation after simplification was \(2z = -2\). To isolate \(z\), we need to eliminate everything that surrounds it. This involves removing the coefficient attached to it by performing the opposite operation. In this context, since the variable \(z\) has a coefficient of \(2\), we will perform division. By doing this, we're essentially undoing any operations that were applied to \(z\).

Just like peeling away layers of an onion, isolating the variable helps you simplify even further until you find the core solution. The final goal is to have the variable on one side and a constant, or simple expression, on the other.

Division is often used in equations to isolate variables. It serves to simplify and uncover the value of the unknown by eliminating coefficients or other numbers attached to it.

In the equation \(2z = -2\), we need to solve for \(z\). Because \(z\) has a coefficient of \(2\), we divide both sides of the equation by \(2\) to "cancel" this multiplying factor. The operation is written as:

• \(z = \frac{-2}{2}\)

Performing the division simplifies the equation to its simplest form: \(z = -1\). This step not only gives you the solution but reinforces the idea that equations are balanced.

Each operation on one side of the equation requires the same operation on the other side. Division, in this context, is a tool that helps balance the equation, leading directly to the answer.