Solving For X In -(2/5)x - 2 = 18 A Step-by-Step Guide

Have you ever stumbled upon an equation that looks a bit intimidating, with fractions and negative signs all over the place? Well, don't worry, guys! We're going to break down one such equation step-by-step and make it super easy to understand. Our mission today is to find the value of x in the equation -(2/5)x - 2 = 18. This might seem like a tough nut to crack, but with a little bit of algebraic magic, we'll have x figured out in no time. So, buckle up, grab your thinking caps, and let's dive into the world of equations!

Understanding the Equation: -(2/5)x - 2 = 18

Before we jump into solving, let's make sure we understand what the equation is telling us. The equation -(2/5)x - 2 = 18 is a linear equation, which means it represents a straight line when graphed. The x is our variable, the unknown value we're trying to find. The fraction -(2/5) is the coefficient of x, and the -2 and 18 are constants. Our goal is to isolate x on one side of the equation, so we can see exactly what it equals. This involves using inverse operations – doing the opposite of what's being done to x. For example, if something is being added to x, we'll subtract it from both sides of the equation. If x is being multiplied by a number, we'll divide both sides by that number. Remember, the golden rule of equation solving is: whatever you do to one side, you must do to the other! This keeps the equation balanced and ensures we get the correct answer. So, let's get started on our algebraic adventure and unravel the mystery of x!

Step 1: Isolating the Term with x

The first step in solving for x is to isolate the term that contains x, which in this case is -(2/5)x. Currently, we have “-2” hanging around on the same side of the equation. To get rid of it, we need to perform the inverse operation. Since 2 is being subtracted, we'll add 2 to both sides of the equation. This is a crucial step because it maintains the balance of the equation. Think of it like a scale – if you add weight to one side, you need to add the same weight to the other side to keep it level. Adding 2 to both sides of -(2/5)x - 2 = 18 gives us: -(2/5)x - 2 + 2 = 18 + 2. Simplifying this, the -2 and +2 on the left side cancel each other out, leaving us with -(2/5)x = 20. Now, we're one step closer to getting x all by itself. This step is all about tidying up the equation and making it easier to work with. By adding 2 to both sides, we've successfully isolated the term with x, which sets us up for the next step: getting rid of that pesky fraction!

Step 2: Eliminating the Fraction

Now that we have -(2/5)x = 20, we need to get rid of the fraction -(2/5) that's multiplying x. The best way to do this is to multiply both sides of the equation by the reciprocal of the fraction. The reciprocal of a fraction is simply flipping it over. So, the reciprocal of -(2/5) is -(5/2). Multiplying by the reciprocal is like undoing the original multiplication. When we multiply a fraction by its reciprocal, we get 1, which is exactly what we want. Multiplying both sides of -(2/5)x = 20 by -(5/2) gives us: -(5/2) * [-(2/5)x] = 20 * -(5/2). On the left side, -(5/2) and -(2/5) cancel each other out, leaving us with just x. On the right side, we have 20 multiplied by -(5/2). To solve this, we can think of 20 as 20/1 and then multiply the numerators and the denominators: (20/1) * -(5/2) = -100/2. Simplifying -100/2 gives us -50. So, our equation now looks like x = -50. We've done it! We've successfully eliminated the fraction and found the value of x.

Step 3: Verifying the Solution

It's always a good idea to check our work, guys, especially when dealing with equations. To verify that x = -50 is the correct solution, we'll plug it back into the original equation -(2/5)x - 2 = 18 and see if it holds true. Substituting x with -50, we get: -(2/5) * (-50) - 2 = 18. First, let's multiply -(2/5) by -50. This gives us (2/5) * 50 = 100/5, which simplifies to 20. So, our equation now looks like: 20 - 2 = 18. Subtracting 2 from 20, we get 18. So, the equation becomes: 18 = 18. This is a true statement, which means our solution x = -50 is indeed correct! Verifying our solution is like the final piece of the puzzle – it gives us confidence that we've solved the equation accurately. It's a step you should never skip, especially on a test or quiz. So, congratulations! We've not only solved for x but also verified our answer.

Final Answer: x = -50

After all the calculations and verifications, we've arrived at our final answer: x = -50. This means that the value of x that satisfies the equation -(2/5)x - 2 = 18 is -50. We've successfully navigated the equation, tackled the fraction, and isolated x. Remember, guys, solving equations is like building a bridge – each step is crucial, and you need to follow the right order to reach the other side. In this case, we first isolated the term with x, then eliminated the fraction, and finally verified our solution. So, the next time you encounter an equation like this, don't be intimidated! Break it down step by step, use inverse operations, and remember the golden rule of equation solving: what you do to one side, you must do to the other. And most importantly, have fun with it! Math can be a fascinating adventure, and solving equations is like cracking a code. So, keep practicing, and you'll become an equation-solving pro in no time!