Simplify Algebraic Expressions: A Math Guide

Understanding the Basics of Algebraic Expressions

Algebraic expressions are the building blocks of much of mathematics, and simplifying them is a fundamental skill. Think of them as mathematical phrases that contain numbers, variables (like 'a', 'b', and 'c'), and operations (addition, subtraction, multiplication, and division). The goal of simplifying is to rewrite an expression in its most concise form, making it easier to understand and work with. This often involves combining like terms, which are terms that have the same variables raised to the same powers. In our specific case, we're dealing with terms that all include the product abc. This makes our task a bit simpler, as all terms are 'like terms' in a sense, differing only by their coefficients (the numbers multiplying the variables). The expression we're looking to simplify is: -((4/3)abc + (1/5)abc) - (-(1/15)abc + (5/12)abc). The presence of parentheses indicates the order of operations. We must first evaluate the expressions within the innermost parentheses before moving outwards. This systematic approach ensures accuracy and avoids common mistakes when dealing with fractions and negative signs.

Step-by-Step Simplification of the Expression

Let's break down the simplification of -((4/3)abc + (1/5)abc) - (-(1/15)abc + (5/12)abc) step by step. Our primary focus is on the coefficients of abc. First, we tackle the terms inside the first set of parentheses: (4/3)abc + (1/5)abc. To add these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 5 is 15. So, we convert 4/3 to (4*5)/(3*5) = 20/15 and 1/5 to (1*3)/(5*3) = 3/15. Adding them gives us (20/15)abc + (3/15)abc = (20+3)/15 * abc = (23/15)abc. Now, this entire first part is multiplied by a negative sign: -(23/15)abc.

Next, we move to the second set of parentheses: -(1/15)abc + (5/12)abc. We have a negative sign in front of (1/15)abc, making it (-1/15)abc. For the second term, (5/12)abc, we need a common denominator for 15 and 12. The LCM of 15 and 12 is 60. We convert -1/15 to (-1*4)/(15*4) = -4/60 and 5/12 to (5*5)/(12*5) = 25/60. Adding these coefficients gives us (-4/60)abc + (25/60)abc = (-4+25)/60 * abc = (21/60)abc. This fraction, 21/60, can be simplified by dividing both numerator and denominator by their greatest common divisor, which is 3. So, 21/60 becomes 7/20. Therefore, the expression inside the second set of parentheses simplifies to (7/20)abc. The original expression now looks like: -(23/15)abc - (7/20)abc.

Finally, we combine these two terms. We have -(23/15)abc and -(7/20)abc. Again, we need a common denominator for 15 and 20. The LCM of 15 and 20 is 60. We convert -23/15 to (-23*4)/(15*4) = -92/60 and -7/20 to (-7*3)/(20*3) = -21/60. Adding these coefficients gives us (-92/60)abc + (-21/60)abc = (-92 - 21)/60 * abc = (-113/60)abc. This fraction, -113/60, cannot be simplified further as 113 is a prime number and is not a factor of 60. Thus, the fully simplified expression is -113/60 * abc.

The Importance of Order of Operations (PEMDAS/BODMAS)

When we simplify algebraic expressions, especially those involving multiple operations and parentheses, adhering to the order of operations is absolutely crucial. This is often remembered by acronyms like PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). Following this established sequence ensures that everyone arrives at the same correct answer. In our expression, -((4/3)abc + (1/5)abc) - (-(1/15)abc + (5/12)abc), the parentheses are our first priority. We must resolve the operations within them before dealing with the negative signs outside or combining terms. The first set of parentheses, (4/3)abc + (1/5)abc, requires us to add fractions. This involves finding a common denominator, which we found to be 15, leading to (23/15)abc. The negative sign outside this group then makes it -(23/15)abc. Similarly, for the second set, (-(1/15)abc + (5/12)abc), we first combine the coefficients inside. The negative sign in front of (1/15)abc means we're effectively adding -1/15 and 5/12. Finding a common denominator of 60, this becomes (-4/60)abc + (25/60)abc = (21/60)abc, which simplifies to (7/20)abc. After resolving the parentheses, the expression becomes -(23/15)abc - (7/20)abc. At this stage, we're left with subtraction of terms with abc. Again, we need a common denominator (60) to combine them: (-92/60)abc - (21/60)abc = (-113/60)abc. Without strictly following PEMDAS/BODMAS, we might incorrectly apply the negative signs or attempt to combine terms prematurely, leading to an incorrect result. Mastering the order of operations is a cornerstone of algebraic manipulation and error prevention.

Dealing with Fractions and Negative Signs in Algebra

Simplifying expressions like the one we've tackled, -((4/3)abc + (1/5)abc) - (-(1/15)abc + (5/12)abc), often involves navigating the complexities of working with fractions and negative signs. These two elements can be tricky if not handled carefully. When adding or subtracting fractions, the fundamental rule is that they must have a common denominator. This means finding the least common multiple (LCM) of the denominators and then adjusting the numerators accordingly. For instance, when we had (4/3)abc + (1/5)abc, the denominators were 3 and 5. Their LCM is 15. So, we converted 4/3 to 20/15 and 1/5 to 3/15, allowing us to add the numerators: (20+3)/15 = 23/15. The variable part abc remains attached. Negative signs introduce another layer of complexity. A negative sign in front of a parenthesis, like - (expression), means you multiply each term inside the parenthesis by -1. This changes the sign of every term within. In our problem, the first part was -(...), so the positive (23/15)abc became -(23/15)abc. The second part involved -(-(1/15)abc + (5/12)abc). The outer negative sign flips the signs of the terms inside. So, -(1/15)abc inside became +(1/15)abc (though we treated it as (-1/15)abc initially and then applied the outer negative sign to the whole sum inside the bracket. More accurately, applying the outer negative sign to -(1/15)abc results in +(1/15)abc and applying it to +(5/12)abc results in -(5/12)abc. However, our step-by-step approach of first simplifying inside the parentheses and then applying the outer sign is often less prone to error. Let's re-examine -( -(1/15)abc + (5/12)abc ). Simplifying inside, (-1/15)abc + (5/12)abc becomes (7/20)abc. Then applying the outer negative sign gives -(7/20)abc. This confirms our earlier calculation. Attention to detail with signs and denominators is key to accurate algebraic simplification.

Conclusion: Mastering Algebraic Simplification

We have successfully simplified the algebraic expression -((4/3)abc + (1/5)abc) - (-(1/15)abc + (5/12)abc) to its most basic form, -113/60 * abc. This process highlights the importance of understanding fundamental algebraic concepts such as combining like terms, finding common denominators for fraction arithmetic, and meticulously applying the order of operations (PEMDAS/BODMAS). Each step, from resolving terms within parentheses to managing negative signs, requires careful attention to detail. Simplifying expressions is not just an academic exercise; it's a critical skill that underpins more complex mathematical problem-solving across various fields, from science and engineering to finance. By consistently practicing these techniques, you build a strong foundation in algebra that will serve you well in your mathematical journey. For further exploration and practice on algebraic manipulation and other math topics, you can visit reliable resources like Khan Academy or Wolfram MathWorld.