Solving For X: A Step-by-Step Guide

Hey guys! Today, we're diving into a common algebraic problem: solving for x. Specifically, we're going to tackle the equation $-15x - 8x - 6x + 16x - 15 = 11$. Don't worry, it might look intimidating at first, but we'll break it down into easy-to-follow steps. By the end of this guide, you'll be a pro at solving similar equations. So, grab your pencils and let's get started!

Understanding the Basics

Before we jump into solving the equation, let's quickly review some basic algebraic concepts. These concepts are the building blocks for solving any equation, so it's crucial to have a firm grasp on them. Think of it like learning the alphabet before you can read – you gotta know the basics!

What is a Variable?

In algebra, a variable is a symbol (usually a letter, like x, y, or z) that represents an unknown value. Our goal when solving an equation is to figure out what that unknown value is. In our case, the variable is x, and we want to find the number that x represents.

What is an Equation?

An equation is a mathematical statement that shows two expressions are equal. It's like a balanced scale – what's on one side must be equal to what's on the other side. Equations always have an equals sign (=). Our equation, $-15x - 8x - 6x + 16x - 15 = 11$, is telling us that the expression on the left side is equal to the number 11 on the right side.

The Goal: Isolating the Variable

The key to solving for x is to isolate it on one side of the equation. This means we want to get x by itself, with no other terms or numbers attached to it. Think of it like giving x its own personal space. To do this, we'll use inverse operations, which we'll talk about in the next section.

Step-by-Step Solution

Okay, now that we've covered the basics, let's get down to the nitty-gritty and solve our equation. We'll go through each step carefully, explaining the reasoning behind it. Remember, practice makes perfect, so don't be afraid to work through this example a few times!

Step 1: Combine Like Terms

The first thing we want to do is simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In our equation, the like terms are the terms with x: -15x, -8x, -6x, and 16x. Let's group them together:

-15x - 8x - 6x + 16x - 15 = 11

Now, we add (or subtract) the coefficients (the numbers in front of the x) of these like terms:

(-15 - 8 - 6 + 16)x - 15 = 11

Simplifying the coefficients, we get:

-13x - 15 = 11

So, the equation now looks much simpler! We've combined all the x terms into a single term.

Step 2: Isolate the Variable Term

Our next goal is to isolate the term with x (-13x) on one side of the equation. To do this, we need to get rid of the -15. Remember the idea of a balanced scale? Whatever we do to one side of the equation, we must do to the other side to keep it balanced.

Since we have -15, we'll add 15 to both sides of the equation. This is the inverse operation of subtraction.

-13x - 15 + 15 = 11 + 15

Simplifying, we get:

-13x = 26

Now, the variable term is isolated on the left side!

Step 3: Solve for x

Finally, we need to isolate x itself. Currently, x is being multiplied by -13. To undo this multiplication, we'll use the inverse operation: division. We'll divide both sides of the equation by -13:

(-13x) / -13 = 26 / -13

Simplifying, we get:

x = -2

And there you have it! We've solved for x. The solution to the equation is x = -2.

Checking Your Solution

It's always a good idea to check your solution to make sure it's correct. This is like proofreading your work before you submit it. To check our solution, we'll substitute x = -2 back into the original equation and see if it holds true.

Original equation: -15x - 8x - 6x + 16x - 15 = 11

Substitute x = -2:

-15(-2) - 8(-2) - 6(-2) + 16(-2) - 15 = 11

Simplify:

30 + 16 + 12 - 32 - 15 = 11

11 = 11

The equation holds true! This confirms that our solution, x = -2, is correct.

Common Mistakes to Avoid

Solving equations can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to watch out for:

• Not combining like terms correctly: Make sure you only combine terms that have the same variable raised to the same power. For example, you can combine -15x and -8x, but you can't combine -15x and -15.
• Not applying operations to both sides: Remember, whatever you do to one side of the equation, you must do to the other side to keep it balanced. If you add 15 to the left side, you must add 15 to the right side as well.
• Incorrectly applying inverse operations: Make sure you use the correct inverse operation to isolate the variable. The inverse of addition is subtraction, and the inverse of multiplication is division.
• Forgetting the sign: Pay close attention to the signs (positive and negative) of the terms. A simple sign error can lead to an incorrect solution.

Practice Makes Perfect

The best way to master solving equations is to practice! Try working through similar problems on your own. The more you practice, the more comfortable you'll become with the steps involved. You can find practice problems in textbooks, online resources, or from your math teacher.

Conclusion

So, there you have it! We've successfully solved the equation $-15x - 8x - 6x + 16x - 15 = 11$ and found that x = -2. Remember, solving for x is a fundamental skill in algebra, and it's essential for tackling more complex problems in the future. By understanding the basics, following the steps carefully, and avoiding common mistakes, you can become a confident equation solver. Keep practicing, and you'll be a pro in no time! And if you have any questions, don't hesitate to ask for help. Happy solving, guys!