Solving 14x - (2x - 2) = 50 A Step-by-Step Algebraic Solution

Hey guys! Today, we're diving into the world of algebra to tackle a fun little equation. We're going to break down the steps to solve the equation 14x - (2x - 2) = 50. Don't worry if it looks intimidating at first; we'll take it one step at a time. Whether you're a student brushing up on your algebra skills or just someone who enjoys a good math puzzle, this guide is for you. We'll cover everything from the basic principles of equation solving to the nitty-gritty details of this specific problem. So, grab a pen and paper, and let's get started!

Understanding the Basics of Algebraic Equations

Before we jump into solving 14x - (2x - 2) = 50, let's quickly review some fundamental concepts of algebraic equations. An algebraic equation is essentially a mathematical statement that shows the equality between two expressions. These expressions can involve numbers, variables (usually represented by letters like 'x'), and mathematical operations such as addition, subtraction, multiplication, and division. The main goal when solving an equation is to isolate the variable on one side of the equation to determine its value. Think of it like a balancing act – whatever you do to one side of the equation, you must do to the other to maintain the balance.

Key principles to remember include the order of operations (PEMDAS/BODMAS), which dictates the sequence in which operations should be performed (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Another crucial concept is the distributive property, which we'll use in this equation to handle the parentheses. The distributive property states that a(b + c) = ab + ac. This means we can multiply a term outside the parentheses by each term inside the parentheses. Understanding these basics will make solving equations like ours much easier and less daunting. Remember, practice makes perfect, so the more you work with these concepts, the more comfortable you'll become.

Step-by-Step Solution: 14x - (2x - 2) = 50

Now, let's get down to business and solve the equation 14x - (2x - 2) = 50 step-by-step. This is where the fun begins! Our main goal is to find the value of 'x' that makes this equation true. To do that, we'll simplify the equation using the principles we discussed earlier and isolate 'x' on one side.

Step 1: Distribute the Negative Sign

The first thing we need to do is tackle those parentheses. Notice that there's a negative sign in front of the parentheses, which means we need to distribute that negative sign to both terms inside the parentheses. It's like multiplying each term inside by -1. So, -(2x - 2) becomes -2x + 2. Remember, a negative times a negative is a positive! Our equation now looks like this: 14x - 2x + 2 = 50.

Step 2: Combine Like Terms

Next up, we need to simplify the left side of the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have 14x and -2x, which are like terms. When we combine them, we simply add their coefficients (the numbers in front of the 'x'). So, 14x - 2x equals 12x. Our equation is now: 12x + 2 = 50.

Step 3: Isolate the Variable Term

Now we want to get the term with 'x' by itself on one side of the equation. To do this, we need to get rid of the + 2 on the left side. We can do this by subtracting 2 from both sides of the equation. Remember, what we do to one side, we must do to the other to keep the equation balanced. So, 12x + 2 - 2 = 50 - 2, which simplifies to 12x = 48.

Step 4: Solve for x

We're almost there! Now we have 12x = 48. To isolate 'x', we need to undo the multiplication. Since 'x' is being multiplied by 12, we'll divide both sides of the equation by 12. So, 12x / 12 = 48 / 12, which simplifies to x = 4. And that's it! We've solved for 'x'.

Step 5: Check Your Solution

It's always a good idea to check your solution to make sure it's correct. To do this, we'll substitute x = 4 back into the original equation and see if it holds true. So, 14(4) - (2(4) - 2) = 50. Let's simplify: 56 - (8 - 2) = 50, then 56 - 6 = 50, and finally 50 = 50. It checks out! This confirms that our solution, x = 4, is indeed correct.

Common Mistakes to Avoid

When solving equations like 14x - (2x - 2) = 50, it's easy to make a few common mistakes. Knowing these pitfalls can help you avoid them and ensure you get the correct answer. One of the most frequent errors is forgetting to distribute the negative sign correctly. Remember, the negative sign in front of the parentheses affects every term inside the parentheses. So, -(2x - 2) should become -2x + 2, not -2x - 2. Another common mistake is combining terms that are not like terms. You can only combine terms that have the same variable raised to the same power. For example, you can combine 14x and -2x, but you can't combine 14x and 2. It's also important to follow the order of operations (PEMDAS/BODMAS) to ensure you're performing operations in the correct sequence. Finally, always double-check your work, especially when dealing with negative signs and fractions. A small error early on can throw off your entire solution. By being mindful of these common mistakes, you can significantly improve your accuracy in solving algebraic equations.

Practice Problems

Now that we've walked through the solution and discussed common mistakes, it's time for some practice! Solving algebraic equations is like learning any new skill – the more you practice, the better you'll become. Here are a few problems similar to 14x - (2x - 2) = 50 that you can try on your own. Remember to follow the same steps we used earlier: distribute, combine like terms, isolate the variable term, solve for x, and check your solution. Feel free to use this guide as a reference if you get stuck, but try to work through the problems independently to build your confidence and skills.

1. Solve for x: 10x - (3x + 5) = 30
2. Solve for y: 8y + (4 - 2y) = 22
3. Solve for z: 15z - (5z - 10) = 60

Working through these problems will solidify your understanding of the process and help you develop a knack for solving equations. Don't be afraid to make mistakes – they're a natural part of learning. The key is to learn from your mistakes and keep practicing. If you get stuck on any of these problems, try revisiting the steps we discussed earlier or seeking out additional resources online or from a tutor. With consistent practice, you'll become a pro at solving algebraic equations in no time!

Real-World Applications of Algebra

You might be wondering,