Evaluating 2p - 3r + T For P=10, R=-4 And T=5 A Step-by-Step Guide

Aug 2, 2025 by Pleton 67 views

Evaluating the Expression 2p - 3r + t: A Comprehensive Guide

Hey guys! Ever stumbled upon an algebraic expression and felt a bit lost? No worries, we've all been there. Today, we're going to break down a common type of problem: evaluating expressions. Specifically, we'll be tackling the expression 2p - 3r + t given the values p = 10, r = -4, and t = 5. By the end of this guide, you'll not only know how to solve this particular problem, but you'll also have a solid understanding of the general process for evaluating any algebraic expression. So, let's dive in and make math a little less intimidating!

Understanding Algebraic Expressions

Before we jump into the solution, let's quickly recap what algebraic expressions are all about. An algebraic expression is a combination of variables (like our p, r, and t), constants (like the numbers 2 and 3 in our expression), and mathematical operations (addition, subtraction, multiplication, division, etc.). The goal of evaluating an expression is to find its numerical value when we know the values of the variables involved. Think of it like a recipe: the expression is the recipe, the variables are the ingredients, and evaluating the expression is like cooking the dish to see what it tastes like. To properly evaluate expressions, we need to know the order of operations.

The order of operations, often remembered by the acronym PEMDAS (or BODMAS in some regions), tells us the sequence in which we should perform mathematical operations. It stands for:

Parentheses (or Brackets)
Exponents (or Orders)
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

This order is crucial because changing the order can drastically change the result. For instance, if we multiply before we do an exponent, we could get the wrong answer. In our expression, we'll primarily be dealing with multiplication, subtraction, and addition, so we'll focus on those aspects of PEMDAS. Remember, multiplication comes before addition and subtraction, so we'll handle the terms 2p and -3r first.

Step-by-Step Solution for 2p - 3r + t

Alright, let's get down to business and solve our problem 2p - 3r + t for p = 10, r = -4, and t = 5. We'll take it one step at a time to make sure everything is crystal clear.

Step 1: Substitution

The first thing we need to do is substitute the given values for the variables in the expression. This means replacing p with 10, r with -4, and t with 5. Our expression then becomes:

2(10) - 3(-4) + 5

See how we've replaced the variables with their corresponding numerical values? This substitution is the foundation of evaluating any algebraic expression. It turns an abstract expression into a concrete numerical calculation. Always double-check your substitutions to ensure you haven't made any errors. A small mistake here can lead to a completely wrong answer. Remember, attention to detail is key in mathematics! Substitution is all about careful and accurate replacement. Once you've mastered this step, the rest is just arithmetic.

Step 2: Multiplication

Now, following the order of operations (PEMDAS), we need to perform the multiplication operations before we tackle the addition and subtraction. We have two multiplication terms in our expression: 2(10) and -3(-4).

Let's start with 2(10). This simply means 2 multiplied by 10, which equals 20. So, we can replace 2(10) with 20 in our expression.

Next, we have -3(-4). Remember that multiplying two negative numbers results in a positive number. So, -3 multiplied by -4 equals 12. We can replace -3(-4) with 12 in our expression.

After performing the multiplication, our expression now looks like this:

20 + 12 + 5

We've successfully eliminated the multiplication operations, making our expression simpler and easier to solve. It's important to pay close attention to signs when multiplying, especially when dealing with negative numbers. A common mistake is to forget the negative sign or to misapply the rules of multiplication with negative numbers. Always double-check your multiplication, paying special attention to the signs.

Step 3: Addition

With the multiplication out of the way, we're left with only addition. Our expression is now 20 + 12 + 5. We can perform the addition from left to right. So we will add 20 and 12 first. Then add the result to 5.

First, let's add 20 and 12. 20 + 12 equals 32. So, we can replace 20 + 12 with 32 in our expression.

Now our expression looks like this:

32 + 5

Next, we add 32 and 5. 32 + 5 equals 37.

Therefore, the final result of our expression is 37. It's crucial to perform the addition in the correct order, especially if there were subtraction operations involved as well. Remember, addition and subtraction have the same precedence in the order of operations, so we perform them from left to right. We've now reached the final answer by carefully following the steps of substitution, multiplication, and addition. This step-by-step approach ensures accuracy and helps to avoid common errors.

Final Answer

So, after carefully following each step, we've arrived at our final answer. The value of the expression 2p - 3r + t when p = 10, r = -4, and t = 5 is 37.

Therefore, 2(10) - 3(-4) + 5 = 20 + 12 + 5 = 37.

We've successfully evaluated the expression by substituting the given values, performing the multiplication, and then carrying out the addition. This process demonstrates the importance of following the order of operations to arrive at the correct result. Remember, evaluating expressions is a fundamental skill in algebra, and mastering it will pave the way for solving more complex problems in the future.

Common Mistakes to Avoid

When evaluating algebraic expressions, there are a few common pitfalls that students often encounter. Being aware of these mistakes can help you avoid them and ensure you get the correct answer. Let's take a look at some of these common errors and how to steer clear of them.

Incorrect Order of Operations

One of the biggest culprits behind incorrect answers is not following the order of operations (PEMDAS/BODMAS). Forgetting to multiply before adding or subtracting, or performing operations in the wrong sequence can lead to significant errors. For example, in our problem, if we had added before multiplying, we would have ended up with a completely different result. To avoid this, always write out each step clearly and double-check that you're following the correct order.

Sign Errors

Dealing with negative numbers can be tricky, and sign errors are a common mistake. Forgetting to include a negative sign or misapplying the rules for multiplying and dividing negative numbers can throw off your entire calculation. Remember that a negative number multiplied by a negative number results in a positive number, and a negative number multiplied by a positive number results in a negative number. In our example, the term -3(-4) requires careful attention to the signs. To minimize sign errors, double-check your signs at each step and consider using parentheses to keep track of negative numbers.

Substitution Errors

Another frequent mistake is incorrectly substituting values for variables. This can happen if you're rushing through the problem or if you're not paying close attention to which value corresponds to which variable. A simple mix-up in substitution can lead to a wrong answer, even if you perform the rest of the calculations correctly. To prevent substitution errors, write down the values of the variables clearly before you start, and double-check your substitutions in the expression. It's also helpful to use different colored pens or highlighters to match variables and their values.

Arithmetic Errors

Even if you understand the concepts and follow the correct steps, simple arithmetic errors can still creep in. Miscalculating a multiplication, addition, or subtraction can lead to an incorrect final answer. These errors are often the result of carelessness or rushing through the calculations. To minimize arithmetic errors, take your time, write out each step clearly, and double-check your calculations. If you're unsure, use a calculator to verify your arithmetic. Remember, accuracy is just as important as understanding the process.

Not Showing Your Work

While it might be tempting to do calculations in your head or skip steps, not showing your work can make it difficult to catch mistakes. When you write out each step clearly, you create a record of your thought process, making it easier to identify any errors you might have made. Additionally, showing your work helps your teacher or instructor understand your approach and give you partial credit even if you make a mistake. Make sure to show every step clearly. It will help you to find and correct your own errors before they get too far.

Practice Problems

Now that we've walked through the solution and discussed common mistakes, it's time to put your knowledge to the test! Practice makes perfect, so let's tackle a few more problems similar to the one we just solved. Working through these examples will solidify your understanding of evaluating algebraic expressions and help you build confidence.

Problem 1

Evaluate the expression 5x + 2y - z for x = 3, y = -2, and z = 4.

Follow the same steps we used in the previous example: substitute the values, perform the multiplication, and then carry out the addition and subtraction. Remember to pay close attention to the signs and the order of operations. Work through this problem on your own, showing each step clearly. Once you've arrived at an answer, compare your solution to the one provided below.

Problem 2

Evaluate the expression -4a - b + 6c for a = -1, b = 5, and c = 2.

This problem involves negative numbers, so be extra careful with your signs. Remember that multiplying two negative numbers results in a positive number, and a negative number multiplied by a positive number results in a negative number. Substitute the values, perform the multiplication, and then carry out the addition and subtraction, working from left to right. Show your work and double-check each step to ensure accuracy.

Problem 3

Evaluate the expression 3(p - q) + 2r for p = 7, q = 2, and r = -3.

This problem introduces parentheses, so remember to follow the order of operations (PEMDAS/BODMAS) and perform the operation inside the parentheses first. Subtract q from p, then multiply the result by 3. Next, multiply 2 by r. Finally, add the two results together. Show your work and pay close attention to the signs.

Solutions

Problem 1:

5(3) + 2(-2) - 4 = 15 - 4 - 4 = 7

Problem 2:

-4(-1) - 5 + 6(2) = 4 - 5 + 12 = 11

Problem 3:

3(7 - 2) + 2(-3) = 3(5) - 6 = 15 - 6 = 9

Conclusion

Great job, guys! You've made it through a comprehensive guide on evaluating the expression 2p - 3r + t. We've covered everything from understanding algebraic expressions and the order of operations to step-by-step solutions, common mistakes, and practice problems. By now, you should have a solid grasp of how to evaluate algebraic expressions accurately and confidently.

Remember, the key to success in math is practice. The more you work through problems, the more comfortable and proficient you'll become. So, keep practicing, keep asking questions, and never give up on your mathematical journey. You've got this! And keep an eye out for our upcoming guides, where we'll tackle even more exciting math topics. Until then, happy calculating!