Evaluate (2/3)b + (1/2)(-a) If A=3, B=-2
Hey guys! Today, we're going to tackle a super fun math problem. We're given an algebraic expression and specific values for the variables, and our mission is to evaluate the expression. Don't worry, it's not as intimidating as it sounds! We'll break it down step by step, making sure everyone understands the process. So, let's dive right in!
Understanding the Problem
Our mission, should we choose to accept it (and we totally do!), is to evaluate the expression (2/3)b + (1/2)(-a). This looks a bit complicated at first glance, but it's really just a matter of substituting the given values for the variables and then doing some arithmetic. We're told that a = 3, b = -2, and c = 4. Notice that 'c' is given a value, but it doesn't actually appear in our expression. This is a common trick in math problems, just to see if you're paying attention! So, we can safely ignore the value of 'c' for this particular problem. The key here is understanding substitution, which is the process of replacing a variable with its given value. Once we substitute, the expression will become a simple arithmetic problem that we can solve using the order of operations.
Breaking Down the Expression
Before we jump into substituting the values, let's take a closer look at the expression itself: (2/3)b + (1/2)(-a). We can see two main parts here, separated by the addition symbol (+). The first part is (2/3)b, which means (2/3) multiplied by b. Remember, in algebra, when a number is written next to a variable without any operation symbol, it implies multiplication. The second part is (1/2)(-a), which means (1/2) multiplied by -a. The parentheses around -a are there to clearly show that the negative sign applies to the variable 'a'. It's crucial to pay attention to these details, as a misplaced sign can completely change the answer. Now that we have a good understanding of the expression, we're ready to move on to the next step: substitution. Remember, substitution is the heart of this problem, so let's make sure we get it right!
Importance of Order of Operations
Before we get into the nitty-gritty of the solution, let's have a quick chat about the order of operations. You might have heard of PEMDAS or BODMAS – it's the golden rule that dictates the order in which we perform mathematical operations. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS is just a slightly different acronym that means the same thing: Brackets, Orders, Division and Multiplication, Addition and Subtraction. The order of operations is like the grammar of mathematics; it ensures that everyone arrives at the same answer for a given expression. If we ignored this order and just performed the operations from left to right, we'd likely end up with the wrong result. So, keep PEMDAS/BODMAS in mind as we work through the problem. It's our trusty guide in the world of mathematical expressions!
Step-by-Step Solution
Okay, let's get down to business and solve this problem step by step. We'll take it nice and slow, making sure we don't miss any details. Remember, the goal is to evaluate (2/3)b + (1/2)(-a) when a = 3 and b = -2.
1. Substitution
The first step, as we discussed earlier, is substitution. We're going to replace the variables 'a' and 'b' with their given values. So, everywhere we see 'a', we'll put 3, and everywhere we see 'b', we'll put -2. This gives us:
(2/3)(-2) + (1/2)(-3)
See how we've replaced 'b' with -2 and 'a' with 3? It's that simple! Now, our expression looks a lot more like a regular arithmetic problem. We've successfully transformed it from an algebraic expression to a numerical one. Substitution is a fundamental skill in algebra, so it's super important to get comfortable with it. It's the key to unlocking many mathematical puzzles. With the variables out of the way, we can now focus on the arithmetic. Remember PEMDAS/BODMAS – it's going to be our guiding light!
2. Multiplication
Now that we've substituted the values, it's time to perform the multiplication operations. According to PEMDAS/BODMAS, multiplication comes before addition, so we need to tackle those multiplications first. We have two multiplications to take care of:
- (2/3)(-2)
- (1/2)(-3)
Let's do them one at a time. First, (2/3)(-2). To multiply a fraction by a whole number, we can think of the whole number as a fraction with a denominator of 1. So, -2 can be written as -2/1. Now we can multiply the fractions: (2/3) * (-2/1) = (2 * -2) / (3 * 1) = -4/3. Remember, a positive number multiplied by a negative number gives a negative result. Now, let's move on to the second multiplication: (1/2)(-3). Again, we can write -3 as -3/1. So, (1/2) * (-3/1) = (1 * -3) / (2 * 1) = -3/2. We've successfully performed both multiplications! Our expression now looks like this:
-4/3 + (-3/2)
We're getting closer to the final answer. All that's left is the addition, but before we can add these fractions, we need to make sure they have a common denominator.
3. Finding a Common Denominator
Before we can add the two fractions, -4/3 and -3/2, we need to find a common denominator. A common denominator is a number that both denominators (3 and 2 in this case) divide into evenly. The easiest way to find a common denominator is to multiply the two denominators together. So, 3 * 2 = 6. Therefore, 6 is our common denominator. Now we need to rewrite each fraction with a denominator of 6. To rewrite -4/3 with a denominator of 6, we multiply both the numerator and the denominator by 2: (-4 * 2) / (3 * 2) = -8/6. To rewrite -3/2 with a denominator of 6, we multiply both the numerator and the denominator by 3: (-3 * 3) / (2 * 3) = -9/6. Now our expression looks like this:
-8/6 + (-9/6)
We're almost there! We've successfully transformed the fractions to have a common denominator, and now we can finally perform the addition.
4. Addition
Now that our fractions have a common denominator, we can add them. To add fractions with a common denominator, we simply add the numerators and keep the denominator the same. So, -8/6 + (-9/6) = (-8 + -9) / 6 = -17/6. And there we have it! We've performed the addition and arrived at our final answer. The expression -17/6 is an improper fraction, meaning the numerator is larger than the denominator. We can leave it like this, or we can convert it to a mixed number. To convert it to a mixed number, we divide 17 by 6. 6 goes into 17 two times with a remainder of 5. So, -17/6 is equal to -2 and 5/6. Both -17/6 and -2 5/6 are perfectly valid answers. The key takeaway here is that we followed the order of operations, performed the necessary arithmetic, and arrived at the solution. We've successfully evaluated the expression!
Final Answer
So, after all that hard work, what's our final answer? We found that when a = 3 and b = -2, the expression (2/3)b + (1/2)(-a) evaluates to -17/6, which is also equal to -2 and 5/6. Yay! We did it! Remember, the most important thing is to understand the process. Math isn't just about getting the right answer; it's about understanding how you got the answer. By breaking down the problem into smaller steps and carefully following the order of operations, we were able to successfully navigate this problem. And that's something to be proud of!
Checking Our Work
It's always a good idea to check our work, especially in math. A simple mistake can sometimes lead to a completely different answer. So, let's quickly review our steps to make sure we haven't made any errors. We started by substituting the values of 'a' and 'b' into the expression. Then, we performed the multiplications, remembering to pay attention to the signs. Next, we found a common denominator for the fractions so we could add them. Finally, we performed the addition and arrived at our answer. If we're confident in each of these steps, then we can be pretty sure our answer is correct. Another way to check our work is to use a calculator. Most calculators can handle fractions and can be used to evaluate expressions like this one. Plugging in the original expression and the values for 'a' and 'b' should give us the same result we calculated by hand. Checking our work is a great habit to develop, as it helps us catch errors and build confidence in our problem-solving abilities.
Conclusion
Alright guys, we've reached the end of our math adventure for today! We successfully evaluated the expression (2/3)b + (1/2)(-a) when a = 3 and b = -2. We learned the importance of substitution, the order of operations (PEMDAS/BODMAS), and how to add fractions with common denominators. More importantly, we learned how to break down a problem into smaller, manageable steps. Remember, math can be challenging, but it's also incredibly rewarding. By practicing regularly and understanding the underlying concepts, you can conquer any math problem that comes your way. So, keep practicing, keep learning, and keep having fun with math! You've got this!
