Solve -120 = 8x: Step-by-Step Solution

Hey guys! Today, we're diving into a fun little algebraic equation: -120 = 8x. Don't worry if it looks intimidating at first. We're going to break it down step-by-step so you can solve it like a pro. Math can seem tricky sometimes, but with a bit of patience and understanding, you’ll realize it's all about following a process. This equation is a classic example of a linear equation, and mastering these basics is super important for more advanced math later on. So, grab your pencils, and let's get started!

Understanding the Equation

First things first, let’s understand what the equation -120 = 8x is telling us. In this equation, -120 is equal to 8 times the value of x. Our main goal here is to figure out what that value of x actually is. The 'x' is what we call a variable, and it represents an unknown number that we're trying to find. Equations like this are the bread and butter of algebra, and they show up everywhere, from simple problems to complex calculations in science and engineering. Understanding the relationship between the numbers and the variable is the first big step in solving the equation. So, remember, we're looking for a number that, when multiplied by 8, gives us -120. Keep that in mind as we move on to the next step!

Algebraic equations are a fundamental part of mathematics, and this particular equation is a simple yet crucial example of a linear equation. The equation -120 = 8x presents a clear relationship: -120 is the result of multiplying 8 by an unknown number, which we represent as 'x.' The beauty of algebra lies in its ability to represent unknown quantities with symbols, allowing us to manipulate and solve for these unknowns using a set of rules and operations. Before we jump into the solution, let’s take a moment to appreciate the structure of this equation. We have a constant (-120) on one side of the equals sign and a term (8x) on the other side. Our mission, should we choose to accept it, is to isolate 'x' on one side of the equation. This means we need to undo the operation that’s currently affecting 'x,' which in this case, is multiplication by 8. Recognizing this fundamental principle of isolating the variable is key to solving not just this equation, but a wide range of algebraic problems. So, with this understanding in place, we're ready to move on to the next step: figuring out how to undo that multiplication and get 'x' all by itself.

Understanding the context of the equation within the broader landscape of mathematics can also be incredibly helpful. Linear equations, like the one we're tackling today, are the foundation upon which much of algebra is built. They represent straight lines when graphed, and their simplicity makes them a great starting point for learning more complex mathematical concepts. But don't let their simplicity fool you – linear equations have real-world applications in fields ranging from economics to physics. For instance, you might use a linear equation to calculate the cost of a service based on an hourly rate, or to predict the distance a car will travel at a constant speed over time. So, by mastering the art of solving equations like -120 = 8x, you're not just learning a mathematical trick; you're building a skill that can be applied in countless practical situations. Now that we've established the importance and context of our equation, let's dive into the specific steps we'll take to find the value of 'x'. Remember, the key is to isolate 'x' by undoing the multiplication, and we'll do that using the inverse operation: division. Stay tuned as we break down the process step-by-step, making it crystal clear how to tackle this type of problem.

The Key: Isolating 'x'

The main strategy to solve for x is to isolate it on one side of the equation. This means we want to get 'x' all by itself, with no other numbers hanging around it on the same side. In our equation, -120 = 8x, the 'x' is currently being multiplied by 8. To undo this multiplication, we need to perform the inverse operation, which is division. Think of it like this: multiplication and division are like opposite sides of a coin. If we divide both sides of the equation by 8, we can cancel out the multiplication and get 'x' by itself. This principle of doing the same thing to both sides of an equation is crucial in algebra. It ensures that the equation remains balanced, like a scale. Whatever you add, subtract, multiply, or divide on one side, you must do the same on the other side to maintain the equality. So, remember, our goal is to get 'x' alone, and we'll do that by dividing both sides by the number that's currently attached to it, which in this case is 8.

Isolating the variable 'x' is the cornerstone of solving algebraic equations. It's like the golden rule of algebra – the fundamental principle that guides us through the process. In essence, isolating 'x' means rearranging the equation so that 'x' is alone on one side, with all other terms on the opposite side. This allows us to clearly see the value of 'x' and know that we've successfully solved the equation. In our case, the equation -120 = 8x has 'x' being multiplied by 8. The key to isolating 'x' is to perform the opposite operation, also known as the inverse operation, on both sides of the equation. This is where division comes in. Division is the inverse of multiplication, and by dividing both sides of the equation by 8, we effectively undo the multiplication that's happening to 'x'. This process might seem simple, but it's incredibly powerful. It's the same technique you'll use to solve a wide variety of algebraic problems, from basic linear equations to more complex scenarios involving multiple variables and operations. The beauty of this method is its consistency and reliability – it works every time, as long as you apply the principle of doing the same thing to both sides of the equation. This ensures that the equation remains balanced, maintaining the equality between the two sides. So, with the concept of isolating 'x' firmly in our minds, let's move on to the next step: actually performing the division and seeing how it works in practice.

The concept of inverse operations is central to isolating variables in algebraic equations. Think of it as a mathematical balancing act – to maintain the equality of an equation, any operation performed on one side must also be performed on the other. In our equation, -120 = 8x, 'x' is being multiplied by 8. To counteract this multiplication and isolate 'x', we employ the inverse operation: division. Just as addition and subtraction are inverse operations, multiplication and division are too. By dividing both sides of the equation by 8, we're essentially