Problem 56 - Elementary Algebra

Question

Solve the conditional equation \(x+6=-2\).

Step-by-Step Solution

Answer

Answer: The solution for the equation is \(x = -8\).

Step 1: Identify the problem

We are given the equation \(x + 6 = -2\). We need to isolate x to find the solution.

Step 2: Subtract 6 from both sides of the equation

In order to isolate x, we need to remove the constant 6 from the equation. Subtract 6 from both sides: \((x + 6) - 6 = -2 - 6\).

Step 3: Simplify the equation

After subtracting 6 from both sides, we get the simplified equation: \(x = -8\).

Step 4: State the solution for x

The solution for the equation is \(x = -8\). This means that when x has the value of -8, the equation is true.

Key Concepts

Isolating Variables

Understanding how to isolate variables is a fundamental skill you'll use in algebra and beyond. The goal is to get the unknown variable—let's call it 'x'—by itself on one side of the equation. Think of it as a balancing act; whatever you do to one side of the equation, you must do to the other to maintain balance.

Imagine you're looking at a scale with equal weights on both sides. If you add or remove weight from one side, you must do the same to the other side to keep it level. This balance is exactly what you're maintaining when you're isolating variables. By systematically using operations like addition, subtraction, multiplication, and division across the equation, 'x' becomes the star of the show, unaccompanied by other numbers or variables.

Subtracting Constants

Subtracting constants from both sides of an equation is a move you'll make often to isolate your variable. A constant is a number that doesn't change, unlike a variable, which can represent many different values. When you face an equation like \(x + 6 = -2\), the number 6 is a constant attached to the variable x.

To get x on its own, you need to subtract this constant from both sides of the equation. Think of it as peeling away what's unnecessary to reveal what's important—like removing the wrapping to find the gift inside. Once you've subtracted the constant from both sides, what's left is a simplified expression leading you one step closer to the solution.

Simplified Equations

A simplified equation is your goalpost when resolving algebraic equations. Simplifying is all about breaking down the complex to make it straightforward. After subtracting constants and performing any necessary arithmetic, you're left with the simplest form of the equation.

For example, when you have the equation \(x + 6 = -2\) and you subtract 6 from both sides, you're simplifying the equation to its essence: \(x = -8\). A simplified equation has all like terms combined, constants subtracted or added, and the variable isolated. It may sound like the end of the journey, but it's actually a beacon: a clear, concise statement that tells you exactly what x should be for the equation to hold true. With a simplified equation, you not only find the solution but also gain a deeper understanding of the structure and relationship within the equation itself.