Solving And Verifying 5x + 18 = 8 A Step-by-Step Guide

Hey there, math enthusiasts! Ever stumbled upon an equation that seems like a puzzle waiting to be solved? Well, today we're diving deep into one such equation: 5x + 18 = 8. This might look intimidating at first glance, but trust me, with a sprinkle of algebra and a dash of logical thinking, we can crack this code together. This article serves as your ultimate guide to not only solving this equation but also verifying the solution to ensure accuracy. So, buckle up and let's embark on this mathematical adventure!

Decoding the Equation: 5x + 18 = 8

Before we jump into solving, let's break down what this equation is telling us. In the realm of algebra, equations are like balanced scales. The goal is to find the value of the unknown, which in this case is represented by the variable 'x'. Our main goal here is to isolate 'x' on one side of the equation to reveal its true value. Think of it as detective work – we're piecing together clues to uncover the hidden number that makes the equation true.

The equation 5x + 18 = 8 is a linear equation, meaning it involves a variable raised to the power of 1. The '5x' part indicates that 'x' is being multiplied by 5, and the '+ 18' tells us that 18 is being added to the result. The '= 8' signifies that the entire expression on the left side of the equation is equal to 8. To solve for 'x', we need to undo these operations in reverse order, maintaining the balance of the equation at all times. Remember, in mathematics, every move we make on one side must be mirrored on the other side to keep the equation honest and true.

Step-by-Step Solution: Unveiling the Value of 'x'

Alright, let's get our hands dirty and solve this equation step by step. First things first, we want to isolate the term with 'x' – that's the '5x'. To do this, we need to get rid of the '+ 18' on the left side. The golden rule of equation solving? Do the opposite! Since we're adding 18, we'll subtract 18 from both sides of the equation. This ensures that the balance is maintained.

So, we have:

5x + 18 - 18 = 8 - 18

This simplifies to:

5x = -10

Now, we're one step closer! We have '5x' on one side, but we want just 'x'. Remember, '5x' means 5 times 'x'. To undo the multiplication, we'll do the opposite operation: division. We'll divide both sides of the equation by 5.

This gives us:

5x / 5 = -10 / 5

Simplifying, we get:

x = -2

Eureka! We've found our solution. The value of 'x' that satisfies the equation is -2. But hold your horses, guys! We're not done yet. It's crucial to verify our solution to make sure we haven't made any sneaky errors along the way.

The Importance of Verification: Ensuring Accuracy

Verification is like the quality control department of equation solving. It's the process of plugging our solution back into the original equation to see if it holds true. Why is this so important? Well, think of it as double-checking your work. Mistakes can happen, and verification is our safety net. It ensures that the value we've found for 'x' truly makes the equation balance. Verification gives us the confidence that our solution is correct and that we can move forward with certainty.

Imagine building a bridge without checking the calculations – that's a recipe for disaster! Similarly, in mathematics, skipping the verification step can lead to incorrect conclusions and wasted effort. It's a small investment of time that yields huge dividends in terms of accuracy and understanding. So, let's embrace verification as an essential part of our problem-solving toolkit.

The Verification Process: Putting Our Solution to the Test

Now for the moment of truth! We've found that x = -2, and now we need to verify if this value truly satisfies our original equation, 5x + 18 = 8. To do this, we'll substitute -2 in place of 'x' in the original equation and see if the left side equals the right side. Remember, the goal is to show that both sides are equal when we plug in our solution. This will confirm that -2 is indeed the correct value for 'x'.

So, let's replace 'x' with -2:

5(-2) + 18 = 8

Now, we'll simplify the left side following the order of operations. First, we multiply 5 by -2:

-10 + 18 = 8

Next, we add -10 and 18:

8 = 8

Boom! The left side of the equation equals the right side. This means that our solution, x = -2, is verified. We've successfully solved the equation and confirmed that our answer is correct. Pat yourselves on the back, guys! This is how we conquer mathematical challenges.

Common Mistakes to Avoid: A Troubleshooting Guide

Solving equations can sometimes feel like navigating a maze, and it's easy to make a wrong turn. But fear not! By being aware of common pitfalls, we can avoid them and stay on the path to success. One frequent mistake is forgetting to apply the same operation to both sides of the equation. Remember, the equation is like a balance – what you do to one side, you must do to the other.

Another common error is mixing up the order of operations. It's crucial to follow the correct sequence (PEMDAS/BODMAS) to ensure accurate calculations. For example, multiplication and division should be done before addition and subtraction. Sign errors are also a frequent culprit. Pay close attention to positive and negative signs, as a small mistake here can throw off the entire solution. Finally, always remember to verify your solution! This simple step can catch many errors before they become a problem.

Real-World Applications: Why This Matters

You might be wondering,