Solving The Equation: -x + 8 + 3x = X - 6
Hey guys! Let's dive into solving this equation together. We've got -x + 8 + 3x = x - 6, and our mission is to find the value of x that makes this equation true. Don't worry, we'll break it down step by step so it’s super clear. Whether you're brushing up on your algebra skills or tackling this for the first time, you're in the right place.
Understanding the Basics
Before we jump into solving, let’s quickly recap some essential concepts. When we solve an equation, we're essentially trying to isolate the variable (in this case, x) on one side of the equals sign. We do this by performing the same operations on both sides of the equation to maintain balance. Think of it like a seesaw – whatever you do on one side, you have to do on the other to keep it level. This includes combining like terms, adding or subtracting values, and multiplying or dividing. Keeping this balance is crucial for finding the correct solution. Remember, the goal is to simplify the equation until we have x = some value. This 'some value' is our answer!
Combining Like Terms
Our first step usually involves combining like terms. Like terms are terms that have the same variable raised to the same power. In our equation, -x and 3x are like terms. We can combine them by simply adding their coefficients. The coefficient is the number in front of the variable. So, in -x, the coefficient is -1, and in 3x, it's 3. Adding these together, we get -1 + 3 = 2. This means we can simplify -x + 3x to 2x. Understanding how to combine like terms is a fundamental skill in algebra, and it’s something you’ll use constantly as you tackle more complex equations. Always look for terms that can be combined; it's a great way to simplify the equation and make it easier to solve.
The Golden Rule of Equations
Remember the golden rule of equations: What you do to one side, you must do to the other. This principle is the backbone of equation solving. It ensures that the equation remains balanced, and the equality holds true. For instance, if we add a number to one side, we must add the same number to the other side. Similarly, if we multiply one side by a number, we must multiply the other side by the same number. This rule applies to all operations – addition, subtraction, multiplication, and division. Keeping the equation balanced is essential for arriving at the correct solution. If you forget this rule, you might end up with an incorrect answer. So, always keep this golden rule in mind as we proceed!
Step-by-Step Solution
Okay, let's get to the fun part – actually solving the equation! We’ll go through each step carefully, explaining the logic behind it. This way, you'll not only get the answer but also understand the process. Trust me, once you get the hang of it, solving equations can be quite satisfying.
Step 1: Combine Like Terms
Looking at our equation, -x + 8 + 3x = x - 6, we can see that -x and 3x are like terms on the left side. Let's combine them. As we discussed earlier, -x is the same as -1x. So, we have -1x + 3x, which simplifies to 2x. Our equation now looks like this: 2x + 8 = x - 6. See how much cleaner that looks already? Combining like terms is a powerful technique for simplifying equations and making them more manageable. It's always a good first step when tackling any algebraic equation.
Step 2: Move Variables to One Side
Next, we want to get all the x terms on one side of the equation. It doesn't matter which side, but it’s often easier to move the smaller x term. In our case, we have 2x on the left and x on the right. To move the x from the right side, we subtract x from both sides. Remember the golden rule? What we do to one side, we do to the other. So, we have:
2x + 8 - x = x - 6 - x
This simplifies to:
x + 8 = -6
Now all our x terms are on the left side. Great progress!
Step 3: Isolate the Variable
We're almost there! Now we need to isolate x completely. This means getting rid of the + 8 on the left side. To do this, we subtract 8 from both sides of the equation:
x + 8 - 8 = -6 - 8
This simplifies to:
x = -14
And there we have it! We’ve solved for x. Our solution is x = -14.
Verification: Checking Our Answer
It's always a good idea to check our answer to make sure we didn't make any mistakes along the way. To do this, we substitute our solution, x = -14, back into the original equation and see if it holds true. This step gives us confidence that our solution is correct.
Our original equation was:
-x + 8 + 3x = x - 6
Substituting x = -14, we get:
-(-14) + 8 + 3(-14) = -14 - 6
Let's simplify both sides:
14 + 8 - 42 = -20
22 - 42 = -20
-20 = -20
It checks out! Both sides of the equation are equal, so our solution x = -14 is correct. Verifying our solution is a crucial step in the problem-solving process. It helps catch any errors and ensures that we have the right answer. So, always take the time to plug your solution back into the original equation to check your work.
Common Mistakes to Avoid
When solving equations, it's easy to make small mistakes that can lead to the wrong answer. Let's look at some common pitfalls to watch out for.
Forgetting the Golden Rule
One of the most common mistakes is forgetting to perform the same operation on both sides of the equation. If you add a number to one side but forget to add it to the other, the equation becomes unbalanced, and your solution will be incorrect. Always remember the golden rule: What you do to one side, you must do to the other.
Incorrectly Combining Like Terms
Another common error is incorrectly combining like terms. For example, you might try to combine terms that don't have the same variable or the same exponent. Remember, like terms must have the same variable raised to the same power. Double-check that you're combining the correct terms.
Sign Errors
Sign errors are easy to make, especially when dealing with negative numbers. Be careful when adding, subtracting, multiplying, or dividing negative numbers. It's a good idea to double-check your signs at each step to avoid these errors. A small sign mistake can throw off the entire solution.
Not Distributing Properly
If the equation involves parentheses, you need to distribute properly. This means multiplying the number outside the parentheses by each term inside the parentheses. Forgetting to distribute or distributing incorrectly is a common mistake. Take your time and make sure you're distributing correctly.
By being aware of these common mistakes, you can avoid them and increase your chances of solving equations correctly. Practice makes perfect, so keep working on these skills, and you'll become a pro at solving equations in no time.
Practice Problems
Want to sharpen your skills? Let’s try a few more practice problems. Working through different types of equations will help you become more confident and proficient.
- Solve for
y:3y - 5 = 7 + y - Solve for
a:2(a + 3) = 4a - 2 - Solve for
z:5z - 8 + 2z = 3z + 4
Take your time, apply the steps we’ve discussed, and remember to check your answers. The key to mastering algebra is practice, practice, practice! Try solving these problems on your own first. If you get stuck, review the steps we covered earlier, and don't hesitate to ask for help. Learning together makes it even better.
Conclusion
So, we've successfully solved the equation -x + 8 + 3x = x - 6, and found that x = -14. We've also covered the fundamental concepts, step-by-step solution, verification, common mistakes, and even some practice problems. Solving equations might seem challenging at first, but with a clear understanding of the basics and consistent practice, you can conquer any algebraic problem. Keep up the great work, and happy solving!