Evaluate Expressions With F = 6: Step-by-Step Guide
Hey guys! Today, we're diving into the world of algebra to tackle a common type of problem: evaluating expressions. Specifically, we'll be working with expressions that involve a variable, f, and we'll be finding their values when f is equal to 6. Don't worry if algebra sounds intimidating – we'll break it down step by step so it's super easy to understand. Let's get started!
Understanding the Basics of Algebraic Expressions
Before we jump into solving the problems, let's quickly recap what algebraic expressions are. An algebraic expression is a combination of numbers, variables (like our f), and mathematical operations (like addition, subtraction, multiplication, and division). The goal when we evaluate expressions is to substitute a given value for the variable and then simplify the expression to find a numerical result.
In our case, the variable is f, and we're given that f = 6. This means that wherever we see f in our expressions, we're going to replace it with the number 6. It’s like giving f a specific identity for this problem. This is a crucial step in understanding how to evaluate algebraic expressions, and mastering this concept will greatly benefit you in more complex algebraic problems. Now that we've got the basics down, let's dive into the first expression.
Part a) Evaluating 5f - 9 + 3f
Step 1: Substitute the value of f
The first expression we need to evaluate is 5f - 9 + 3f. Remember, the first thing we do is substitute f with its given value, which is 6. So, we replace every f in the expression with 6. This gives us:
5(6) - 9 + 3(6)
See? It's just like replacing a placeholder. This step is super important because it sets the stage for the rest of the calculation. Without this initial substitution, we can't move forward. Substituting correctly is the foundation of evaluating expressions in algebra, and paying close attention to this step will minimize errors down the line.
Step 2: Perform the multiplications
Next, we need to take care of the multiplications. Remember the order of operations (PEMDAS/BODMAS)? Multiplication comes before addition and subtraction. We have two multiplications to perform:
- 5(6) = 30
- 3(6) = 18
Now our expression looks like this:
30 - 9 + 18
Multiplication is a key operation here because it simplifies the expression by removing the variables and turning them into numerical values. Think of it as scaling up the value of f according to the coefficients in front of it. Performing these multiplications correctly is vital for solving algebraic problems, and it's a skill that will be used extensively in more advanced math topics.
Step 3: Add and subtract from left to right
Now we're left with addition and subtraction. We perform these operations from left to right. First, we subtract 9 from 30:
30 - 9 = 21
Then, we add 18 to the result:
21 + 18 = 39
So, the value of the expression 5f - 9 + 3f when f = 6 is 39. Yay! We solved the first one. This step is all about combining the numerical values we've calculated so far. Following the left-to-right rule ensures we adhere to standard mathematical conventions. Mastering this process is essential for accurate algebraic calculations, and it’s a skill that’s applicable beyond just this type of problem.
Summary of Part a)
To recap, we substituted f with 6, performed the multiplications, and then did the addition and subtraction from left to right. The final value of the expression 5f - 9 + 3f when f = 6 is 39. Remember these steps, guys – they're your toolkit for tackling similar problems. Understanding the logical flow of these operations is key to mastering algebra. Now that we’ve successfully tackled our first expression, let’s move on to the second one, which involves a fraction. Don’t worry, we’ll break it down just as easily!
Part b) Evaluating (4f - 3) / 7
Step 1: Substitute the value of f
Just like before, the first step is to substitute f with its value, which is 6. We're dealing with the expression (4f - 3) / 7. Replacing f with 6 gives us:
(4(6) - 3) / 7
Notice how the entire expression 4f - 3 is inside parentheses? This is important because it tells us to perform the operations inside the parentheses before we do the division. Parentheses act like a VIP section in the order of operations, so anything inside them gets priority. This initial substitution is a critical step, and it’s the same process we used in the first part of the problem. Ensuring you correctly substitute the value of the variable sets the stage for accurate evaluation of algebraic expressions.
Step 2: Perform the multiplication inside the parentheses
Inside the parentheses, we have 4(6) - 3. According to the order of operations, we need to do the multiplication first:
4(6) = 24
Now our expression inside the parentheses looks like this:
24 - 3
Remember, the order of operations is like a set of rules that everyone agrees on in math. It ensures that we all get the same answer when we solve a problem. Multiplication gets the green light before subtraction in this case. This step is crucial because it simplifies the expression within the parentheses, preparing us for the next operation. Understanding and applying the order of operations is a cornerstone of algebraic problem-solving, and it’s something you’ll use repeatedly.
Step 3: Perform the subtraction inside the parentheses
Now we subtract 3 from 24:
24 - 3 = 21
Our expression now looks like this:
21 / 7
We've simplified the expression inside the parentheses to a single number. This is a big step forward! By performing the subtraction, we’re one step closer to getting our final answer. Subtraction is a fundamental arithmetic operation, and it's essential to perform it accurately to correctly evaluate algebraic expressions. This step highlights the importance of following the order of operations to maintain accuracy in our calculations.
Step 4: Perform the division
Finally, we divide 21 by 7:
21 / 7 = 3
So, the value of the expression (4f - 3) / 7 when f = 6 is 3. Awesome! We've solved the second part of the problem. Division is the final operation in this particular expression, and it brings us to the final numerical value. By performing this division, we’re completing the process of simplifying the original algebraic expression. This step demonstrates how each operation plays a crucial role in evaluating expressions, and it emphasizes the importance of understanding the relationships between different mathematical operations.
Summary of Part b)
We substituted f with 6, performed the multiplication inside the parentheses, then the subtraction, and finally, the division. The value of the expression (4f - 3) / 7 when f = 6 is 3. Great job, guys! Remember how we tackled the parentheses first? That's key! This step-by-step approach is what makes solving algebraic expressions manageable and straightforward. By breaking down the problem into smaller, more digestible parts, we can avoid confusion and ensure accuracy.
Conclusion: Mastering Algebraic Evaluation
And there you have it! We've successfully evaluated two algebraic expressions for f = 6. Remember, the key is to substitute the variable with its given value and then follow the order of operations (PEMDAS/BODMAS). By breaking the problem down into smaller steps, like substitution, multiplication, addition, subtraction, and division, you can confidently tackle any algebraic expression that comes your way.
Understanding how to evaluate algebraic expressions is a fundamental skill in algebra, and it's something that will come up again and again in your math journey. So, practice these steps, guys, and you'll become algebraic whizzes in no time! Keep practicing, and you’ll find that these concepts become second nature. The more comfortable you are with these basics, the easier it will be to tackle more complex algebraic problems in the future. Keep up the great work, and remember to have fun with math!