Evaluating Algebraic Expressions With X=1 And Y=-1
Hey guys! Today, we're diving into the world of algebra and tackling the challenge of evaluating expressions when we're given specific values for our variables. It might sound intimidating, but trust me, it's totally doable, and we'll break it down step-by-step. We'll be working with two expressions, and our mission is to find their values when x equals 1 and y equals -1. So, let's put on our mathematical thinking caps and get started!
(i) Evaluating the Expression $5x - 3y + 5$
Okay, our first expression is $5x - 3y + 5$. The key to evaluating this lies in substitution. We're going to replace the variables, x and y, with their given values. Remember, x is 1 and y is -1. So, let's rewrite the expression with these values plugged in. We get $5(1) - 3(-1) + 5$. See how we've swapped out the letters with the numbers? Now, it's just a matter of following the order of operations (PEMDAS/BODMAS) to simplify. Multiplication comes first, so we have 5 times 1, which is 5, and -3 times -1, which is +3 (remember, a negative times a negative is a positive!). Our expression now looks like this: $5 + 3 + 5$.
Next up, we tackle the addition. We simply add the numbers together: 5 plus 3 equals 8, and 8 plus 5 equals 13. So, the value of the expression $5x - 3y + 5$ when x = 1 and y = -1 is 13. Boom! We've successfully evaluated our first expression. It's like solving a puzzle, where each step brings us closer to the final answer. This process of substituting and simplifying is fundamental in algebra, and mastering it opens the door to solving more complex equations and problems. Think of it as building a strong foundation for your mathematical journey. The more you practice, the more comfortable and confident you'll become in handling these types of problems. Remember, the key is to take it one step at a time, carefully substituting the values and following the order of operations. You've got this!
(ii) Evaluating the Expression $\frac{3x}{2} - \frac{3y}{6}$
Alright, let's move on to our second expression: $\frac{3x}{2} - \frac{3y}{6}$. This one involves fractions, but don't let that scare you! We'll approach it in the same systematic way as before. Our first step, as always, is substitution. We'll replace x with 1 and y with -1. This gives us $\frac{3(1)}{2} - \frac{3(-1)}{6}$. Notice how we've carefully placed the values into the expression, making sure to maintain the correct positions and operations. Now, let's simplify each fraction separately.
For the first fraction, $\frac3(1)}{2}$, we multiply 3 by 1, which equals 3. So, the first fraction becomes $\frac{3}{2}$. For the second fraction, $\frac{3(-1)}{6}$, we multiply 3 by -1, which equals -3. This gives us $\frac{-3}{6}$. Now our expression looks like this{2} - \frac{-3}{6}$. Aha! We have a subtraction of a negative, which is the same as addition. So, we can rewrite the expression as $\frac{3}{2} + \frac{3}{6}$. Before we can add these fractions, we need a common denominator. The least common multiple of 2 and 6 is 6. So, we'll convert $\frac{3}{2}$ to an equivalent fraction with a denominator of 6. To do this, we multiply both the numerator and the denominator of $\frac{3}{2}$ by 3, which gives us $\frac{9}{6}$.
Now our expression is $\frac9}{6} + \frac{3}{6}$. We can now add the numerators, keeping the denominator the same{6} = \frac{12}{6}$. Finally, we simplify the fraction $\frac{12}{6}$, which equals 2. Therefore, the value of the expression $\frac{3x}{2} - \frac{3y}{6}$ when x = 1 and y = -1 is 2. Fantastic! We've successfully tackled an expression with fractions. Remember, the key here was to break the problem down into smaller, manageable steps: substitute, simplify each fraction, find a common denominator, add, and then simplify the final result. By following this process, you can conquer even the trickiest algebraic expressions. Keep practicing, and you'll become a fraction-busting pro in no time!
Key Takeaways and Tips for Evaluating Expressions
So, guys, we've successfully navigated through two examples of evaluating algebraic expressions. Before we wrap up, let's recap some key takeaways and helpful tips that will boost your expression-evaluating skills. These are the nuggets of wisdom that will make the process smoother and more efficient. Think of them as your secret weapons in the world of algebra.
First and foremost, substitution is your best friend. This is the cornerstone of evaluating expressions. Always start by carefully substituting the given values for the variables. Double-check that you've placed the values in the correct spots and that you haven't missed any negative signs. A small error in substitution can throw off your entire calculation, so take your time and be meticulous. It's like laying the foundation for a building – a strong foundation ensures a sturdy structure.
Next up, master the order of operations. Remember PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is the golden rule of simplifying expressions. Following the correct order ensures that you perform operations in the right sequence, leading to the correct answer. Think of it as following a recipe – each step needs to be done in the correct order to achieve the desired result. A slight deviation can change the outcome, so keep PEMDAS/BODMAS close at heart.
When dealing with fractions, finding a common denominator is crucial. Before you can add or subtract fractions, they need to have the same denominator. This allows you to combine the numerators while keeping the denominator consistent. Remember, the least common multiple (LCM) is your go-to for finding the common denominator. It's like speaking the same language – fractions need to have a common base to communicate effectively. Once you've mastered finding common denominators, fraction operations become a breeze.
Don't forget the power of simplification. After performing operations, always check if you can simplify your answer further. This might involve reducing fractions to their simplest form or combining like terms. Simplification not only gives you the most concise answer but also makes it easier to work with the result in future calculations. It's like polishing a gem – simplification brings out the brilliance of your solution.
Finally, practice makes perfect. The more you practice evaluating expressions, the more comfortable and confident you'll become. Start with simple expressions and gradually work your way up to more complex ones. Don't be afraid to make mistakes – they're valuable learning opportunities. Each problem you solve strengthens your understanding and hones your skills. It's like building muscle memory – the more you exercise, the stronger you become. So, keep practicing, and you'll be an expression-evaluating whiz in no time!
By keeping these tips in mind, you'll be well-equipped to tackle any algebraic expression that comes your way. So, go forth and conquer the world of algebra, one expression at a time! Remember, math is a journey, not a destination. Enjoy the process, embrace the challenges, and celebrate your successes. You've got this!
Conclusion
In conclusion, evaluating algebraic expressions with given values is a fundamental skill in mathematics. By understanding the principles of substitution, order of operations, and simplification, you can confidently tackle these types of problems. We've seen how to break down complex expressions into manageable steps, and we've learned valuable tips for ensuring accuracy and efficiency. Remember, the key is to practice consistently and to approach each problem with a systematic mindset. With dedication and the right strategies, you can master the art of evaluating expressions and unlock new levels of mathematical understanding. So, keep exploring, keep learning, and keep pushing your mathematical boundaries. The world of algebra awaits your expertise!
