Simplifying Algebraic Expressions: A Step-by-Step Guide To 120 - (10 + 2)y + 6 - 1

by ADMIN 83 views

Hey guys! Today, we're diving into a simple yet essential math problem: simplifying algebraic expressions. This is a fundamental skill that you'll use constantly in algebra and beyond, so let's break it down step by step. We're tackling the expression 120 - (10 + 2)y + 6 - 1, and I promise, it's not as intimidating as it looks! Our goal is to make this expression as clean and easy to work with as possible. Think of it like decluttering your room – you want to organize everything neatly so you can find what you need quickly.

Understanding the Order of Operations

Before we jump into the nitty-gritty, let's quickly recap the order of operations, often remembered by the acronym PEMDAS (or BODMAS in some parts of the world). This is our golden rule for simplifying expressions, and it tells us the sequence in which we should perform operations:

  1. Parentheses (or Brackets)
  2. Exponents (or Orders)
  3. Multiplication and Division (from left to right)
  4. Addition and Subtraction (from left to right)

Following this order ensures that we all arrive at the same correct answer. Imagine if we didn't have PEMDAS – we could end up with a bunch of different answers for the same problem! So, PEMDAS is our trusty guide in the world of math.

Let's Apply PEMDAS to Our Expression

Now, let's bring PEMDAS into action with our expression: 120 - (10 + 2)y + 6 - 1. The first thing we spot is the parentheses, so that's where we'll begin. Inside the parentheses, we have (10 + 2), which is a straightforward addition. We add 10 and 2 to get 12. So, we've simplified the parentheses part to 12. Our expression now looks like this: 120 - 12y + 6 - 1. See? We're already making progress!

Next up, we look for exponents, but we don't have any in this expression. So, we move on to multiplication and division. In our updated expression, 120 - 12y + 6 - 1, we see 12y, which means 12 multiplied by y. We can't simplify this further right now because y is a variable, an unknown quantity. So, we leave 12y as it is and move on to the next operation.

Now we tackle addition and subtraction. We have three numbers to combine: 120, +6, and -1. Remember, we perform these operations from left to right. First, let's add 120 and 6. 120 plus 6 equals 126. So now we have 126 - 1. Finally, we subtract 1 from 126, which gives us 125. Great! We've simplified the numerical part of our expression. Now, let’s put it all together.

Putting It All Together: The Simplified Expression

After following PEMDAS, we've simplified the original expression 120 - (10 + 2)y + 6 - 1 step by step. We first dealt with the parentheses, then handled the multiplication (the term with y), and finally combined the constants through addition and subtraction. So, let's recap what we've done.

We started with 120 - (10 + 2)y + 6 - 1. We simplified (10 + 2) to 12, giving us 120 - 12y + 6 - 1. We couldn't simplify 12y further, so we moved on to the addition and subtraction. We combined 120, 6, and -1 to get 125. Now, we combine the terms we've simplified to get our final expression.

The simplified expression is 125 - 12y. This is the most straightforward form of our original expression. We've combined all the like terms, leaving us with a constant term (125) and a term with the variable y (-12y). This simplified form is much easier to work with if we were to, say, solve an equation or evaluate the expression for a specific value of y.

Why Simplification Matters

You might be wondering, “Why go through all this trouble to simplify an expression?” Well, simplification is a crucial skill in algebra for several reasons. First, simplified expressions are easier to understand and interpret. Looking at 125 - 12y is much clearer than looking at 120 - (10 + 2)y + 6 - 1. The simplified form allows us to quickly see the relationship between the variables and constants in the expression.

Second, simplification makes it easier to solve equations. If you had an equation like 120 - (10 + 2)y + 6 - 1 = 0, you'd want to simplify the left side first to 125 - 12y = 0 before trying to solve for y. This makes the solving process much more manageable.

Third, simplified expressions are less prone to errors. The more terms and operations you have, the higher the chance of making a mistake. By simplifying, you reduce the number of steps and thus the opportunity for errors.

Practice Makes Perfect

Like any skill, simplifying expressions becomes easier with practice. The more you do it, the more comfortable you'll become with the process. Start with simple expressions and gradually work your way up to more complex ones. Try varying the types of operations and the number of terms to challenge yourself. And remember, PEMDAS is your best friend in this journey. Keep it handy, and you'll be simplifying expressions like a pro in no time!

Real-World Applications

Simplifying expressions isn't just an abstract math concept; it has many real-world applications. For example, in physics, you might need to simplify an equation to calculate the trajectory of a projectile. In finance, you might simplify a formula to determine the total cost of a loan. In computer programming, simplified code is often more efficient and easier to debug. The ability to simplify expressions is a valuable skill in many fields.

Moreover, simplifying expressions helps in developing logical thinking and problem-solving skills. It teaches you to break down complex problems into smaller, manageable steps. This approach is useful not just in math but in many areas of life. When faced with a challenging situation, you can apply the same principles of simplification: identify the core components, break down the problem into smaller parts, and solve each part step by step.

Common Mistakes to Avoid

When simplifying expressions, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them. One common mistake is not following the order of operations correctly. For instance, someone might add before multiplying, which would lead to an incorrect answer. Always double-check that you're following PEMDAS.

Another mistake is mishandling negative signs. It's easy to make a mistake when distributing a negative sign across parentheses or when combining negative numbers. Take your time and be careful with the signs. A simple way to avoid errors is to rewrite subtraction as addition of a negative number. For example, a - b can be rewritten as a + (-b).

Lastly, some students make mistakes when combining like terms. Remember, you can only combine terms that have the same variable and exponent. For example, 3x and 5x are like terms and can be combined to get 8x, but 3x and 5x² are not like terms and cannot be combined. Pay close attention to the variables and exponents when combining terms.

Tips for Success

To excel at simplifying expressions, here are some tips that can help:

  1. Write neatly: Clear handwriting can prevent many errors. When your work is organized, it's easier to spot mistakes and keep track of your steps.
  2. Show your work: Don't try to do too much in your head. Write down each step, so you can easily review your work and identify any errors.
  3. Check your answer: After simplifying an expression, take a moment to check your work. You can do this by plugging in a value for the variable and comparing the result of the original expression with the simplified expression. If they match, you're likely on the right track.
  4. Practice regularly: Consistent practice is the key to mastering any math skill. Set aside some time each day or week to work on simplifying expressions. The more you practice, the more confident you'll become.
  5. Seek help when needed: If you're struggling with simplifying expressions, don't hesitate to ask for help. Talk to your teacher, a tutor, or a classmate. Explaining your difficulties can often help you understand the concepts better.

Wrapping Up

So, we've journeyed through simplifying the expression 120 - (10 + 2)y + 6 - 1, and hopefully, you've picked up some valuable insights along the way. Remember, guys, simplifying expressions is a core skill in algebra, and mastering it opens doors to more advanced math concepts. By following the order of operations (PEMDAS), being mindful of common mistakes, and practicing regularly, you can become a simplification whiz!

We started by understanding the importance of PEMDAS, then applied it step by step to our expression. We simplified the parentheses, dealt with multiplication, and combined like terms through addition and subtraction. We saw why simplification matters, from making expressions easier to understand to reducing errors in calculations. We also discussed real-world applications of simplification and common mistakes to avoid.

Keep practicing, stay curious, and don't be afraid to tackle challenging problems. Math is a journey, and every step you take, every problem you solve, brings you closer to your goal. You've got this!