Simplify $5(x+6\frac{1}{2})-(x+9\frac{1}{3})-(19-x)$ A Step-by-Step Guide

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In the realm of mathematics, simplifying expressions is a fundamental skill. It's the cornerstone of solving equations, understanding complex concepts, and building a solid foundation for advanced topics. In this article, we will embark on a step-by-step journey to simplify the expression $5(x + 6\frac{1}{2}) - (x + 9\frac{1}{3}) - (19 - x)$. We'll break down each step, explain the underlying principles, and provide valuable insights to help you master this essential skill. Whether you're a student grappling with algebra or a lifelong learner seeking to brush up on your math skills, this guide will equip you with the knowledge and confidence to tackle any simplification challenge.

Understanding the Order of Operations

Before we dive into the specifics of our expression, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS or BODMAS. This set of rules dictates the sequence in which mathematical operations should be performed:

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

By adhering to the order of operations, we ensure that we arrive at the correct simplified form of the expression. Failing to follow this order can lead to errors and incorrect results. Think of it as the grammar of mathematics – just as proper grammar ensures clear communication in language, the order of operations ensures accurate calculations in math.

Step 1: Distribute the 5

The first part of our expression is $5(x + 6\frac{1}{2})$. According to PEMDAS, we need to address the parentheses first. However, we can't directly add $x$ and $6\frac{1}{2}$ because they are different terms. This is where the distributive property comes into play. The distributive property states that $a(b + c) = ab + ac$. In other words, we multiply the term outside the parentheses by each term inside the parentheses.

Applying the distributive property to our expression, we get:

$5(x + 6\frac{1}{2}) = 5 * x + 5 * 6\frac{1}{2}$

Now, let's simplify further. $5 * x$ is simply $5x$. To multiply 5 by $6\frac{1}{2}$, it's helpful to convert the mixed number to an improper fraction. $6\frac{1}{2}$ is equal to $\frac{13}{2}$ (because 6 * 2 + 1 = 13, and we keep the same denominator). So, we have:

$5 * \frac{13}{2} = \frac{5 * 13}{2} = \frac{65}{2}$

We can leave this as an improper fraction or convert it back to a mixed number, which is $32\frac{1}{2}$. Therefore, the first part of our expression simplifies to:

$5x + 32\frac{1}{2}$

Mastering the distributive property is key to simplifying expressions, especially those involving parentheses. It allows you to break down complex expressions into more manageable parts. Remember, the goal is to eliminate the parentheses while maintaining the expression's value.

Step 2: Distribute the Negative Sign

Next, we have $-(x + 9\frac{1}{3})$. This might seem straightforward, but it's a common place for errors. Remember that the negative sign in front of the parentheses is equivalent to multiplying the entire expression inside the parentheses by -1. This is another application of the distributive property, but with a negative multiplier.

So, we distribute the -1 to both terms inside the parentheses:

$-(x + 9\frac{1}{3}) = -1 * x + (-1) * 9\frac{1}{3} = -x - 9\frac{1}{3}$

Again, to work with the mixed number, we convert $9\frac{1}{3}$ to an improper fraction: $9\frac{1}{3} = \frac{28}{3}$ (because 9 * 3 + 1 = 28). So, we have:

$-x - \frac{28}{3}$

This step highlights the importance of paying close attention to signs in mathematical expressions. A misplaced negative sign can completely alter the result. Always treat a negative sign in front of parentheses as a multiplication by -1.

Step 3: Distribute the Negative Sign (Again)

We encounter another set of parentheses with a negative sign in front: $-(19 - x)$. We apply the same principle as in the previous step, distributing the -1 to both terms inside the parentheses:

$-(19 - x) = -1 * 19 + (-1) * (-x) = -19 + x$

Notice that multiplying -1 by -x results in +x. This is a crucial rule to remember: a negative times a negative is a positive. This step reinforces the importance of careful sign manipulation in algebraic expressions.

Step 4: Combine Like Terms

Now that we've eliminated all the parentheses, our expression looks like this:

$5x + 32\frac{1}{2} - x - 9\frac{1}{3} - 19 + x$

The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, the like terms are: $5x$, $-x$, and $x$ (these are the 'x' terms), and $32\frac{1}{2}$, $-9\frac{1}{3}$, and $-19$ (these are the constant terms).

Let's group the like terms together:

$(5x - x + x) + (32\frac{1}{2} - 9\frac{1}{3} - 19)$

Now, we can combine the 'x' terms: $5x - x + x = 5x$. Remember that $-x$ is the same as $-1x$ and $x$ is the same as $1x$. So, we're essentially doing 5 - 1 + 1, which equals 5. Therefore, the 'x' terms simplify to $5x$.

Combining like terms is a fundamental skill in algebra. It allows you to simplify expressions and make them easier to work with. Think of it as tidying up your algebraic workspace – grouping similar items together makes the whole process more organized and efficient.

Step 5: Simplify the Constant Terms

Now, let's tackle the constant terms: $32\frac{1}{2} - 9\frac{1}{3} - 19$. This involves subtracting mixed numbers and integers. It's often helpful to convert the mixed numbers to improper fractions to make the calculations easier.

We already know that $32\frac{1}{2} = \frac{65}{2}$ and $9\frac{1}{3} = \frac{28}{3}$. So, our expression becomes:

$\frac{65}{2} - \frac{28}{3} - 19$

To subtract fractions, they need a common denominator. The least common multiple of 2 and 3 is 6. So, we convert the fractions:

$\frac{65}{2} = \frac{65 * 3}{2 * 3} = \frac{195}{6}$

$\frac{28}{3} = \frac{28 * 2}{3 * 2} = \frac{56}{6}$

We also need to express 19 as a fraction with a denominator of 6: $19 = \frac{19 * 6}{6} = \frac{114}{6}$.

Now, we can subtract the fractions:

$\frac{195}{6} - \frac{56}{6} - \frac{114}{6} = \frac{195 - 56 - 114}{6} = \frac{25}{6}$

We can leave this as an improper fraction or convert it back to a mixed number: $\frac{25}{6} = 4\frac{1}{6}$.

Working with fractions can be challenging, but mastering the process of finding common denominators and performing operations is essential for simplifying expressions. Remember to always convert mixed numbers to improper fractions when adding or subtracting.

The Final Simplified Expression

Putting it all together, we have simplified the expression to:

$5x + 4\frac{1}{6}$

This is the simplest form of the expression $5(x + 6\frac{1}{2}) - (x + 9\frac{1}{3}) - (19 - x)$. We have successfully navigated through the order of operations, distributed terms, combined like terms, and simplified fractions to arrive at this final result.

Conclusion

Simplifying expressions is a crucial skill in mathematics, and by following a systematic approach and understanding the underlying principles, you can confidently tackle any simplification challenge. Remember the order of operations (PEMDAS/BODMAS), the distributive property, and the importance of combining like terms. Practice is key to mastering this skill, so work through various examples and gradually increase the complexity of the expressions you tackle. With dedication and the knowledge gained from this guide, you'll be well-equipped to simplify expressions with confidence and excel in your mathematical endeavors.