Solving $100 = -(x-1) + 4(x-6)$: A Step-by-Step Guide

Hey guys! Ever stumbled upon an equation that looks like a tangled mess of numbers and variables? Well, today, we're going to unravel one such mystery: 100 = -(x-1) + 4(x-6). This equation might seem daunting at first glance, but trust me, with a step-by-step approach and a sprinkle of mathematical magic, we can conquer it together. So, buckle up, grab your pencils, and let's dive into the exciting world of solving equations!

Breaking Down the Equation: A Step-by-Step Guide

Before we jump into solving, let's take a moment to understand what this equation is all about. At its core, it's a statement that says the expression on the left side (100) is equal to the expression on the right side (-(x-1) + 4(x-6)). Our mission, should we choose to accept it, is to find the value of 'x' that makes this statement true. This involves simplifying the equation and isolating 'x' on one side.

Step 1: Distribute Like a Pro

The first thing we need to tackle is the parentheses. To do this, we'll use the distributive property, which basically means multiplying the terms outside the parentheses by each term inside. Let's start with the -(x-1) part. Remember, the negative sign in front of the parentheses is like multiplying by -1. So, we have:

-1 * x = -x

-1 * -1 = +1

This simplifies the first part to -x + 1. Now, let's move on to the 4(x-6) part:

4 * x = 4x

4 * -6 = -24

This simplifies the second part to 4x - 24. So, after distributing, our equation now looks like this:

100 = -x + 1 + 4x - 24

Step 2: Combine Like Terms: The Art of Grouping

Now that we've freed ourselves from the parentheses, it's time to tidy things up by combining like terms. Like terms are those that have the same variable raised to the same power. In our equation, we have two 'x' terms (-x and 4x) and two constant terms (1 and -24). Let's group them together:

(-x + 4x) + (1 - 24)

Combining -x and 4x gives us 3x. Combining 1 and -24 gives us -23. So, our equation simplifies further to:

100 = 3x - 23

Step 3: Isolate the Variable: Getting 'x' Alone

Our goal is to get 'x' all by itself on one side of the equation. To do this, we need to get rid of the -23 that's hanging out with the 3x. We can do this by adding 23 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to maintain the balance:

100 + 23 = 3x - 23 + 23

This simplifies to:

123 = 3x

Step 4: Divide and Conquer: The Final Showdown

We're almost there! Now we have 3x = 123. To finally isolate 'x', we need to get rid of the 3 that's multiplying it. We can do this by dividing both sides of the equation by 3:

123 / 3 = 3x / 3

This gives us:

41 = x

So, the solution to our equation is x = 41! We've successfully unlocked the mystery and found the value of 'x' that makes the equation true.

Verifying the Solution: Double-Checking Our Work

To be absolutely sure we've got the right answer, it's always a good idea to plug our solution back into the original equation and see if it holds true. Let's substitute x = 41 into our original equation:

100 = -(41-1) + 4(41-6)

Now, let's simplify:

100 = -(40) + 4(35)

100 = -40 + 140

100 = 100

Ta-da! The equation holds true. This confirms that our solution, x = 41, is indeed correct. It's like a victory dance for our mathematical brains!

Common Pitfalls and How to Avoid Them

Solving equations can be tricky, and there are a few common pitfalls that students often fall into. Let's take a look at some of these and how to avoid them:

• Forgetting to Distribute the Negative Sign: When dealing with a negative sign in front of parentheses, it's crucial to distribute it to every term inside. For example, in -(x-1), the negative sign applies to both the 'x' and the '-1'. Failing to do so can lead to incorrect results.
• Combining Unlike Terms: Only like terms can be combined. You can't add an 'x' term to a constant term, for instance. Make sure you're only combining terms that have the same variable raised to the same power.
• Not Performing the Same Operation on Both Sides: The golden rule of equation solving is that whatever you do to one side, you must do to the other. This maintains the balance of the equation. If you add a number to one side, you have to add it to the other. If you divide one side by a number, you have to divide the other side by the same number.
• Order of Operations Errors: Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Following the correct order is essential for accurate simplification.

By being aware of these common pitfalls and practicing diligently, you can significantly improve your equation-solving skills.

Real-World Applications: Where Equations Come to Life

You might be wondering,