Simplifying Algebraic Expressions A Step-by-Step Guide

In this article, we will thoroughly walk through the process of simplifying the given algebraic expression. Algebraic expressions can often appear complex, but by systematically applying the distributive property and combining like terms, we can reduce them to a simpler, more manageable form. This skill is fundamental in algebra and is crucial for solving equations and understanding more advanced mathematical concepts. Let’s delve into the step-by-step simplification of the expression: $3${\frac{7}{5} x+4}$-2${\frac{3}{2}-\frac{5}{4} x}$.

Understanding the Expression

Before diving into the simplification process, it's essential to understand the structure of the expression. The expression $3${\frac{7}{5} x+4}$-2${\frac{3}{2}-\frac{5}{4} x}$ involves two main parts, each contained within parentheses and multiplied by a constant. The first part, $3${\frac{7}{5} x+4}$, means that the entire expression inside the parentheses, which is $\frac{7}{5} x+4$, is multiplied by 3. Similarly, the second part, $-2${\frac{3}{2}-\frac{5}{4} x}$, indicates that the expression $\frac{3}{2}-\frac{5}{4} x$ is multiplied by -2. The negative sign in front of the 2 is critical and must be distributed correctly to ensure the correct simplification.

The expression contains both terms with the variable x and constant terms (numbers without variables). To simplify, we will first use the distributive property to remove the parentheses. This involves multiplying each term inside the parentheses by the constant outside. After applying the distributive property, we will have a series of terms, some with x and some without. The next step is to combine like terms, which means adding or subtracting terms that have the same variable and exponent (in this case, terms with x) and adding or subtracting the constant terms.

Simplifying algebraic expressions is not just a mechanical process; it’s a fundamental skill that allows us to solve equations, understand functions, and model real-world situations mathematically. A clear understanding of each step ensures accuracy and builds a solid foundation for more advanced algebraic manipulations. By mastering these techniques, you'll be better equipped to tackle complex problems in mathematics and related fields.

Step 1: Apply the Distributive Property

The distributive property is the cornerstone of simplifying expressions that contain parentheses. This property states that for any numbers a, b, and c, a( b + c ) = a b + a c. In simpler terms, it means we multiply the term outside the parentheses by each term inside the parentheses.

Let’s apply this to our expression, $3${\frac{7}{5} x+4}$-2${\frac{3}{2}-\frac{5}{4} x}$. We have two distributions to perform:

1. Distribute the 3 across the terms inside the first set of parentheses: $3 * \frac{7}{5}x + 3 * 4$
2. Distribute the -2 across the terms inside the second set of parentheses: $-2 * \frac{3}{2} - 2 *(-\frac{5}{4} x)$

When performing these multiplications, it is crucial to pay close attention to the signs. A negative number multiplied by a negative number results in a positive number, while a negative number multiplied by a positive number results in a negative number. This is a common area for errors, so careful attention here is vital.

After applying the distributive property, we get:

• $3 * \frac{7}{5}x = \frac{21}{5}x$
• $3 * 4 = 12$
• $-2 * \frac{3}{2} = -3$
• $-2 *(-\frac{5}{4} x) = \frac{10}{4}x$

So, the expression now looks like this: $\frac{21}{5}x + 12 - 3 + \frac{10}{4}x$. Each term inside the original parentheses has been correctly multiplied by its respective constant outside the parentheses. The next step will involve simplifying the fractions and combining the like terms, but this initial distribution is the most critical part of simplifying the expression correctly. If this step is done accurately, the rest of the process becomes much more straightforward.

Step 2: Simplify Fractions and Combine Like Terms

After applying the distributive property, our expression looks like this: $\frac{21}{5}x + 12 - 3 + \frac{10}{4}x$. The next crucial step is to simplify the fractions and then combine like terms. Like terms are terms that contain the same variable raised to the same power (in this case, x) or constant terms (numbers without variables).

First, let’s simplify the fraction $\frac{10}{4}x$. Both 10 and 4 are divisible by 2, so we can reduce the fraction: $\frac{10}{4}x = \frac{5}{2}x$.

Now our expression is: $\frac{21}{5}x + 12 - 3 + \frac{5}{2}x$.

Next, we combine the terms with x. To do this, we need a common denominator for the fractions $\frac{21}{5}$ and $\frac{5}{2}$. The least common denominator (LCD) of 5 and 2 is 10. We convert each fraction to have this denominator:

• $\frac{21}{5}x = \frac{21 * 2}{5 * 2}x = \frac{42}{10}x$
• $\frac{5}{2}x = \frac{5 * 5}{2 * 5}x = \frac{25}{10}x$

Now we can add these terms: $\frac{42}{10}x + \frac{25}{10}x = \frac{42 + 25}{10}x = \frac{67}{10}x$.

Next, we combine the constant terms: $12 - 3 = 9$.

So, after combining like terms, our simplified expression is $\frac{67}{10}x + 9$. This streamlined form is much easier to work with in further calculations or when solving equations. Simplifying fractions and accurately combining like terms are essential algebraic skills that ensure accuracy and efficiency in mathematical problem-solving. By mastering these techniques, you will be well-prepared to tackle more complex algebraic challenges.

Step 3: Final Simplified Expression

After meticulously applying the distributive property, simplifying fractions, and combining like terms, we have arrived at the final simplified form of the expression. Our step-by-step process has led us from the initial complex expression, $3${\frac{7}{5} x+4}$-2${\frac{3}{2}-\frac{5}{4} x}$, to a much more manageable and clear form.

We initially distributed the constants outside the parentheses, which gave us $\frac{21}{5}x + 12 - 3 + \frac{10}{4}x$. Then, we simplified the fraction $\frac{10}{4}x$ to $\frac{5}{2}x$. Following this, we combined the x terms by finding a common denominator and adding the fractions, resulting in $\frac{67}{10}x$. Finally, we combined the constant terms, $12 - 3$, which simplified to 9.

Therefore, the final simplified expression is $\frac{67}{10}x + 9$. This result matches option B in the original question. Simplifying expressions not only makes them easier to understand but also reduces the chances of making errors in subsequent calculations. This process demonstrates the power and efficiency of algebraic manipulation. By breaking down a complex problem into smaller, manageable steps, we can systematically arrive at the correct solution.

The ability to simplify algebraic expressions is a cornerstone of mathematical proficiency. It is essential for solving equations, working with functions, and tackling more advanced topics in mathematics. Each step, from distributing constants to combining like terms, requires careful attention to detail and a solid understanding of the underlying mathematical principles. By mastering these techniques, students and practitioners alike can confidently approach complex algebraic problems and achieve accurate results.

Conclusion

In summary, simplifying algebraic expressions involves a systematic approach that combines the distributive property, simplification of fractions, and the combining of like terms. Throughout this article, we have meticulously walked through the steps required to simplify the expression $3${\frac{7}{5} x+4}$-2${\frac{3}{2}-\frac{5}{4} x}$. Starting with the initial application of the distributive property, we expanded the expression by multiplying each term inside the parentheses by the constants outside, ensuring that we paid close attention to the signs of each term. This step is crucial as it sets the stage for the rest of the simplification process.

Following the distribution, we addressed the fractions within the expression. Simplifying fractions not only makes the numbers easier to work with but also reduces the complexity of the expression as a whole. We identified and simplified $\frac{10}{4}x$ to $\frac{5}{2}x$, which streamlined the subsequent steps. The next critical phase involved combining like terms. This required us to add the terms with the variable x together and the constant terms separately. To add the fractional coefficients of x, we found a common denominator, which allowed us to accurately combine $\frac{21}{5}x$ and $\frac{5}{2}x$. Similarly, we combined the constant terms by performing simple arithmetic.

The final result, $\frac{67}{10}x + 9$, represents the simplified form of the original expression and corresponds to option B. This outcome underscores the importance of each step in the simplification process. A mistake in any one step could lead to an incorrect final answer. Therefore, careful attention to detail and a solid understanding of algebraic principles are essential. Mastering the simplification of algebraic expressions is a foundational skill in mathematics. It not only helps in solving equations and inequalities but also prepares learners for more advanced topics such as calculus and linear algebra. The ability to break down complex expressions into simpler forms is invaluable in both academic and practical settings. By following a systematic approach, one can confidently tackle even the most challenging algebraic expressions.

Therefore, the correct answer is B. $\frac{67}{10} x+9$.