Simplifying Algebraic Expressions 3C-D When C=p-1 And D=-3p+1
Given C = p - 1 and D = -3p + 1, what is the simplified expression for 3C - D in standard form?
In the realm of mathematics, algebraic expressions serve as a fundamental building block for more complex equations and problem-solving. Often, we encounter expressions that require simplification to reveal their underlying structure and make them easier to work with. This article delves into the process of simplifying an algebraic expression, specifically focusing on the expression 3C - D, given that C = p - 1 and D = -3p + 1. Through a step-by-step approach, we will explore how to substitute the given values, distribute constants, combine like terms, and ultimately arrive at the expression in its standard form. This process not only enhances our understanding of algebraic manipulation but also equips us with valuable skills applicable in various mathematical contexts. The ability to simplify expressions efficiently is crucial in solving equations, analyzing functions, and tackling more advanced mathematical concepts. So, let's embark on this journey of algebraic simplification and uncover the elegance and precision inherent in mathematical expressions.
Substituting the Given Values
The first step in simplifying the expression 3C - D is to substitute the given values of C and D. We are given that C = p - 1 and D = -3p + 1. Substituting these values into the expression 3C - D, we get:
3C - D = 3(p - 1) - (-3p + 1)
This substitution is a critical step as it replaces the variables C and D with their respective algebraic equivalents, allowing us to work with a single variable, p, and simplify the expression further. The parentheses in the expression are crucial as they indicate that the entire expression within them should be treated as a single term. This is particularly important when we have constants or negative signs multiplying the expressions, as we will see in the next step. By carefully substituting the given values, we set the stage for the subsequent steps of distribution and combining like terms, which will ultimately lead us to the simplified form of the expression. This foundational step underscores the importance of precision and attention to detail in algebraic manipulations, ensuring that we accurately represent the relationships between variables and constants.
Distributing Constants
After substituting the values of C and D, the next step involves distributing the constants. In our expression, 3(p - 1) - (-3p + 1), we need to distribute the 3 across the terms inside the first parenthesis and the negative sign (which can be thought of as -1) across the terms inside the second parenthesis. Distributing the 3 across (p - 1) gives us 3 * p - 3 * 1, which simplifies to 3p - 3. Next, we distribute the negative sign across (-3p + 1), which means multiplying each term inside the parenthesis by -1. This gives us -1 * -3p + -1 * 1, which simplifies to 3p - 1. So, after distributing the constants, our expression becomes:
3p - 3 + 3p - 1
This step is crucial because it eliminates the parentheses and allows us to combine like terms in the subsequent step. The distributive property is a fundamental concept in algebra, allowing us to simplify expressions that involve multiplication over addition or subtraction. By correctly distributing the constants, we ensure that each term is properly accounted for, and we avoid errors in our simplification process. This step highlights the importance of understanding and applying algebraic properties to manipulate expressions effectively. The ability to distribute constants accurately is a key skill in algebra, enabling us to simplify complex expressions and solve equations with confidence.
Combining Like Terms
Following the distribution of constants, the next crucial step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, 3p - 3 + 3p - 1, we identify the terms with the variable 'p' (which are 3p and 3p) and the constant terms (which are -3 and -1). To combine like terms, we simply add their coefficients. For the 'p' terms, we have 3p + 3p, which combines to 6p. For the constant terms, we have -3 - 1, which combines to -4. Therefore, after combining like terms, our expression simplifies to:
6p - 4
This step is essential because it reduces the number of terms in the expression, making it more concise and easier to understand. Combining like terms is a fundamental algebraic technique that simplifies expressions and prepares them for further manipulation or evaluation. The ability to identify and combine like terms accurately is a critical skill in algebra, as it is used extensively in solving equations, simplifying expressions, and working with polynomials. By combining like terms, we streamline the expression, making it easier to work with and interpret. This process not only simplifies the expression but also reveals its underlying structure, making it easier to analyze and use in further calculations or problem-solving.
Standard Form
After combining like terms, we have arrived at the simplified expression 6p - 4. This expression is now in standard form, which means that the terms are arranged in descending order of their exponents. In this case, we have a term with the variable 'p' (which has an exponent of 1) and a constant term. The term with the highest exponent (6p) is written first, followed by the constant term (-4). Standard form is important because it provides a consistent and organized way of representing algebraic expressions, making them easier to compare and manipulate. An expression in standard form allows for quick identification of the leading coefficient (in this case, 6) and the degree of the expression (which is 1, as 'p' is raised to the power of 1). This form is particularly useful when working with polynomials, as it facilitates operations such as addition, subtraction, multiplication, and division. By expressing the simplified expression in standard form, we ensure clarity and ease of interpretation, which are essential in mathematical communication and problem-solving. The ability to represent expressions in standard form is a key skill in algebra, enabling us to work with algebraic expressions efficiently and effectively.
In conclusion, we have successfully simplified the algebraic expression 3C - D, given that C = p - 1 and D = -3p + 1. Through a series of steps, we demonstrated the process of substituting given values, distributing constants, and combining like terms. Starting with the initial expression, we substituted the values of C and D, resulting in 3(p - 1) - (-3p + 1). We then distributed the constants, which involved multiplying the terms inside the parentheses by the constants outside, leading to 3p - 3 + 3p - 1. Next, we combined like terms, grouping together the terms with the variable 'p' and the constant terms, resulting in 6p - 4. Finally, we expressed the simplified expression in standard form, which is 6p - 4, where the terms are arranged in descending order of their exponents. This final form is a concise and organized representation of the original expression, making it easier to understand and work with. The ability to simplify algebraic expressions is a fundamental skill in mathematics, with applications in various areas, including equation solving, function analysis, and more advanced mathematical concepts. By mastering these techniques, we can confidently tackle more complex problems and gain a deeper understanding of mathematical relationships.
Algebraic expressions, simplify, substitution, distribution, combining like terms, standard form, variables, constants, coefficients, exponents, equations, mathematical problem-solving.