Finding The Sum Of 12-5l And -3+4l A Step-by-Step Guide

Have you ever stumbled upon an algebraic expression that seemed like a puzzle waiting to be solved? Well, you're not alone! Many students and math enthusiasts find themselves scratching their heads when faced with expressions involving variables and constants. But fear not, because today, we're going to break down a fascinating problem step by step. We'll explore how to find the sum of two expressions: $12 - 5l$ and $-3 + 4l$. Buckle up, guys, because we're about to embark on a mathematical adventure!

Understanding the Basics of Algebraic Expressions

Before we dive into the problem at hand, let's quickly recap the fundamental concepts of algebraic expressions. An algebraic expression is a combination of variables, constants, and mathematical operations. Variables are symbols (usually letters) that represent unknown values, while constants are fixed numerical values. Mathematical operations like addition, subtraction, multiplication, and division connect these elements to form meaningful expressions. In our case, the expressions $12 - 5l$ and $-3 + 4l$ contain the variable $l$ and constants like 12, -5, -3, and 4. The key to simplifying and solving these expressions lies in understanding how to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, $3x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1. Similarly, $7y^2$ and $-2y^2$ are like terms. However, $2x$ and $2x^2$ are not like terms because the variable $x$ is raised to different powers. When adding or subtracting algebraic expressions, we can only combine like terms. This means we can add or subtract the coefficients (the numerical part) of the like terms while keeping the variable and its exponent the same. For instance, $3x + 5x = 8x$, and $7y^2 - 2y^2 = 5y^2$. Now that we've refreshed our understanding of algebraic expressions and like terms, we're well-equipped to tackle the problem of finding the sum of $12 - 5l$ and $-3 + 4l$.

Step-by-Step Solution: Adding the Expressions

Alright, let's get down to business and solve this problem! Our mission is to find the sum of $12 - 5l$ and $-3 + 4l$. The first step is to write down the expression representing the sum: $(12 - 5l) + (-3 + 4l)$. Now, to simplify this expression, we need to combine like terms. Remember, like terms are those that have the same variable raised to the same power. In this case, we have two types of terms: constant terms (numbers without variables) and terms with the variable $l$. The constant terms are 12 and -3, and the terms with $l$ are $-5l$ and $4l$. Let's start by combining the constant terms: $12 + (-3)$. This is simply 12 minus 3, which equals 9. So, the constant part of our sum is 9. Next, let's combine the terms with $l$: $-5l + 4l$. To do this, we add the coefficients of $l$, which are -5 and 4. Adding -5 and 4 gives us -1. Therefore, $-5l + 4l = -1l$, which we can also write as $-l$. Now we have all the pieces of the puzzle! We found that the sum of the constant terms is 9, and the sum of the $l$ terms is $-l$. To get the final answer, we simply combine these two results: $9 + (-l)$, which we can write more concisely as $9 - l$. And there you have it! The sum of $12 - 5l$ and $-3 + 4l$ is $9 - l$. It's like putting together a jigsaw puzzle, where each term is a piece that fits perfectly into the final solution. Now, let's take a look at the answer choices provided in the problem and see which one matches our result.

Identifying the Correct Answer Choice

Now that we've diligently worked through the problem and found that the sum of $12 - 5l$ and $-3 + 4l$ is $9 - l$, it's time to match our solution with the answer choices provided. This step is crucial because it ensures that we've not only performed the calculations correctly but also interpreted the question accurately. Let's revisit the answer choices: A. $-16 + 63i$ B. $9 - 1$ C. $9 - 9i$ D. $15 - 9i$. Looking at these options, we can immediately eliminate A, C, and D. Why? Because our answer, $9 - l$, is in the form of a constant minus a variable term, whereas options A, C, and D involve the imaginary unit $i$. The imaginary unit $i$ is defined as the square root of -1, and it's used in complex numbers, which have the form $a + bi$, where $a$ and $b$ are real numbers. Our original expressions and our calculated sum do not involve any imaginary units, so we can confidently rule out these choices. This leaves us with option B: $9 - 1$. At first glance, this might seem different from our answer, $9 - l$. However, if we look closely, we realize that option B is simply a numerical expression that can be further simplified. $9 - 1$ equals 8. So, option B represents the value 8. Now, this is where it gets interesting. Our answer, $9 - l$, is an algebraic expression, while option B is a numerical value. To determine if option B is the correct answer, we need to consider whether the problem implies a specific value for $l$. If $l$ were equal to 1, then our answer, $9 - l$, would indeed be equal to $9 - 1$, which is 8. However, the problem does not provide any information about the value of $l$. It simply asks for the sum of the two expressions in their algebraic form. Therefore, while $9 - 1$ is a valid numerical expression, it doesn't represent the sum of the given expressions in the same way that $9 - l$ does. The correct answer should be $9-l$, but this option is not available, so there may be a typo in the options. If l is 1, then the correct answer is B.

Common Mistakes and How to Avoid Them

In the world of mathematics, it's not uncommon to encounter pitfalls along the way. Adding algebraic expressions is a fundamental skill, but even seasoned mathematicians can make mistakes if they're not careful. Let's explore some common errors that students often make when adding expressions like $12 - 5l$ and $-3 + 4l$, and more importantly, let's learn how to avoid them. One of the most frequent mistakes is failing to combine like terms correctly. As we discussed earlier, like terms are those that have the same variable raised to the same power. For example, $3x$ and $5x$ are like terms, but $2x$ and $2x^2$ are not. When adding expressions, it's crucial to identify and group like terms together. A common error is to mistakenly add or subtract terms that are not alike. For instance, someone might try to combine the constant term 12 with the term $-5l$, which is incorrect. To avoid this, always double-check that you're only combining terms that have the same variable and exponent. Another potential pitfall is mishandling the signs of the terms. Remember that subtraction can be thought of as adding a negative number. So, when you have an expression like $12 - 5l$, it's helpful to think of it as $12 + (-5l)$. This can prevent errors when you're combining terms with negative coefficients. For example, when adding $-5l$ and $4l$, be sure to pay attention to the negative sign in front of the 5. The correct sum is $-l$, not $9l$. A third common mistake is not distributing the addition or subtraction sign correctly when dealing with parentheses. In our problem, we have $(12 - 5l) + (-3 + 4l)$. Since we're adding the expressions, the parentheses don't change the signs of the terms inside. However, if we were subtracting the expressions, we would need to distribute the negative sign to each term in the second expression. For example, if we had $(12 - 5l) - (-3 + 4l)$, we would need to rewrite it as $12 - 5l + 3 - 4l$ before combining like terms. By being mindful of these common mistakes and practicing regularly, you can build your confidence and accuracy in adding algebraic expressions.

Real-World Applications of Algebraic Expressions

Now that we've mastered the art of adding algebraic expressions, you might be wondering,