Simplifying Expressions: A Step-by-Step Guide

Oct 17, 2025 by ADMIN 46 views

Hey everyone! Today, we're diving into the world of simplifying and evaluating algebraic expressions. This might sound a bit intimidating at first, but trust me, it's like solving a fun puzzle! We'll be working through the expression ${\{1 \frac{44}{49} - 11.3 y^{4}}$ + ${6 y^{2} + 11.3 y^{4}}$ , where ${y = -\frac{3}{7}}$ . Let's break it down into manageable chunks and make sure we understand it well. I'll walk you through each step, making sure that it is easy to understand. We'll start by simplifying the expression, which means combining like terms and making it look as clean as possible. Then, we'll substitute the value of ${y}$ and calculate the final answer. Ready? Let's get started!

Step 1: Simplifying the Expression – Combining Like Terms

Alright, guys, our first goal is to simplify the expression. Simplifying expressions is a fundamental skill in algebra. The expression we're dealing with is ${\{1 \frac{44}{49} - 11.3 y^{4}}$ + ${6 y^{2} + 11.3 y^{4}}$ ), and it might look a bit messy right now, but don't worry, we can make it cleaner! The key here is to identify and combine like terms. Like terms are terms that have the same variable raised to the same power.

Looking at our expression, we can see a few types of terms: a constant term ${\{1 \frac{44}{49}}$ , a term with ${y^{4}}$ and another term with ${y^{2}}$ . Notice that we have ${-11.3y^{4}}$ and ${ + 11.3 y^{4}}$ . These are like terms because they both have the ${y^{4}}$ variable. Also, notice that they have opposite signs. When you add these terms, they cancel each other out: ${-11.3 y^{4} + 11.3 y^{4} = 0}$ . In other words, they eliminate each other from the equation. Now, we are left with ${1 \frac{44}{49}}$ and ${6y^{2}}$ . The expression now becomes, ${1 \frac{44}{49} + 6y^{2}}$ . In summary, we've simplified our original expression by combining the ${y^{4}}$ terms, which canceled each other out. This gives us a much simpler expression to work with. Remember, the goal of simplifying is to make the expression easier to handle and understand.

Now, our expression is quite simplified, which brings us to the next step, which will involve evaluating the expression. Remember, in the previous step, we looked at how to simplify by combining like terms and reducing the expression as much as possible. With the skills in that section, you will be able to do this type of math easily.

Converting the Mixed Fraction

Before we move on, let's convert the mixed fraction ${1 \frac{44}{49}}$ to an improper fraction. This will make it easier to work with later. To do this, we multiply the whole number part (1) by the denominator (49) and add the numerator (44). So, ${1 \times 49 + 44 = 49 + 44 = 93}$ . The denominator stays the same, so we get ${\frac{93}{49}}$ . Now, our simplified expression looks like this: ${\frac{93}{49} + 6y^{2}}$ .

Step 2: Evaluating the Expression – Substituting the Value of y

Okay, team, now it's time to evaluate the expression. This means we're going to substitute the given value of ${y}$ into our simplified expression and calculate the result. We've simplified the expression to ${\frac{93}{49} + 6y^{2}}$ , and we're given that ${y = -\frac{3}{7}}$ .

So, wherever we see ${y}$ , we'll replace it with ${-\frac{3}{7}}$ . Doing so, our expression becomes ${\frac{93}{49} + 6(-\frac{3}{7})^{2}}$ . Let's be extra careful with this negative sign. When we square a negative number, the result is positive. Now, let's calculate ${(-\frac{3}{7})^{2}}$ . This means ${(-\frac{3}{7}) \times (-\frac{3}{7})}$ , which equals ${\frac{9}{49}}$ . Our expression now becomes ${\frac{93}{49} + 6 \times \frac{9}{49}}$ . The next step is to multiply ${6}$ by ${\frac{9}{49}}$ . You can think of ${6}$ as ${\frac{6}{1}}$ , so we multiply the numerators (6 and 9) and the denominators (1 and 49). That gives us ${\frac{6 \times 9}{1 \times 49} = \frac{54}{49}}$ . Thus our expression is ${\frac{93}{49} + \frac{54}{49}}$ . Lastly, we add the two fractions, which have the same denominator. ${\frac{93 + 54}{49} = \frac{147}{49}}$ .

Simplifying the Fraction

Let's simplify this fraction! We have ${\frac{147}{49}}$ . We can see that both the numerator and the denominator are divisible by 49. Dividing both by 49, we get ${\frac{147 \div 49}{49 \div 49} = \frac{3}{1}}$ , or simply ${3}$ . Therefore, when we evaluate the expression ${\{1 \frac{44}{49} - 11.3 y^{4}}$ + ${6 y^{2} + 11.3 y^{4}}$ , where ${y = -\frac{3}{7}}$ , the final answer is ${3}$ . Awesome! You've successfully simplified and evaluated the expression! Give yourself a pat on the back.

Step 3: Recap and Key Takeaways

Alright, guys, let's quickly recap what we did and extract some key takeaways.

We started with a complex expression, ${\{1 \frac{44}{49} - 11.3 y^{4}}$ + ${6 y^{2} + 11.3 y^{4}}$ , and simplified it by combining like terms. This step is all about making the expression cleaner and easier to work with. Remember, like terms have the same variable raised to the same power.
We converted the mixed fraction to an improper fraction.
Then, we substituted the given value of ${y = -\frac{3}{7}}$ into the simplified expression, which was ${\frac{93}{49} + 6y^{2}}$ . Evaluating means replacing the variables with their values and performing the calculations.
After the substitution, we followed the order of operations (PEMDAS/BODMAS): parentheses/brackets, exponents/orders, multiplication and division (from left to right), and addition and subtraction (from left to right).
We performed the calculations, being careful with the negative signs and fractions, and found the final answer to be ${3}$ .

Key Takeaways: The core of simplifying and evaluating expressions lies in identifying like terms, substituting the given values, and following the order of operations. Remember that practice makes perfect. The more you work through these types of problems, the easier and more intuitive it will become. Don't be afraid to break down the problem into smaller steps. Double-check your work, especially when dealing with negative signs and fractions. And most importantly, have fun with it! Math is like a puzzle, and it's super rewarding when you solve it.

Bonus Tips and Tricks

Here are some bonus tips and tricks to make simplifying and evaluating expressions even easier:

Practice, Practice, Practice: The more problems you solve, the more comfortable you'll become with the process. Try different types of expressions and vary the values you substitute.
Use the Order of Operations: Always follow the order of operations (PEMDAS/BODMAS) to ensure you perform the calculations in the correct sequence. This will help you avoid common mistakes.
Simplify as You Go: Simplify each step before moving on. This can prevent the numbers from becoming too large and make the calculations easier. For example, if you see a fraction that can be simplified, do it immediately.
Rewrite the Expression: Sometimes, rewriting the expression can help you visualize it more clearly. For instance, if you have terms with negative coefficients, rewrite them with the subtraction sign. This can help prevent errors.
Check Your Work: Always double-check your work, especially when dealing with negative numbers, fractions, and exponents. A small mistake can lead to a wrong answer, so take the time to review your steps.

Conclusion

Great job, everyone! We've successfully simplified and evaluated the expression ${\{1 \frac{44}{49} - 11.3 y^{4}}$ + ${6 y^{2} + 11.3 y^{4}}$ , where ${y = -\frac{3}{7}}$ . We've seen how to combine like terms, substitute values, and follow the order of operations to arrive at a solution. This process builds a strong foundation for more complex algebraic concepts. Keep practicing, and you'll become a pro in no time! Remember to always break down problems into smaller, manageable steps and don't be afraid to ask for help if you need it. Math can be a lot of fun when you approach it with the right mindset. Keep up the excellent work, and I'll see you in the next lesson!