Simplifying Algebraic Expressions A Step-by-Step Guide To -p + 7(5 + 5p)

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In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex equations and reduce them to their most basic, understandable forms. This article delves into the process of simplifying the algebraic expression -p + 7(5 + 5p). We will explore the steps involved, from applying the distributive property to combining like terms, ensuring a clear and comprehensive understanding of the simplification process.

Before we begin the simplification process, let's break down the expression -p + 7(5 + 5p). This expression consists of several components:

  • -p: This is a term with a variable p and a coefficient of -1. It represents the negative value of the variable p.
  • 7(5 + 5p): This part involves multiplication of a constant (7) with a binomial expression (5 + 5p). This is where the distributive property comes into play, as we will need to multiply 7 with each term inside the parentheses.
  • 5: This is a constant term, a numerical value that does not change.
  • 5p: This is another term with the variable p and a coefficient of 5. It represents five times the value of p.

The presence of parentheses and multiple terms indicates that simplification is necessary to make the expression more concise and easier to work with. The process involves applying algebraic principles to eliminate parentheses, combine like terms, and present the expression in its simplest form. Understanding the individual components and their relationships is crucial for a successful simplification.

The first step in simplifying the expression -p + 7(5 + 5p) is to apply the distributive property. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. In other words, we multiply the term outside the parentheses by each term inside the parentheses.

In our expression, the term outside the parentheses is 7, and the terms inside are 5 and 5p. Applying the distributive property, we get:

7 * 5 = 35

7 * 5p = 35p

So, 7(5 + 5p) becomes 35 + 35p. Now, we can rewrite the original expression as:

-p + 35 + 35p

This step is crucial because it eliminates the parentheses, allowing us to combine like terms in the next step. The distributive property is a fundamental tool in algebra, and mastering its application is essential for simplifying expressions and solving equations. By correctly applying this property, we transform the expression into a form where like terms can be easily identified and combined.

After applying the distributive property, our expression is now -p + 35 + 35p. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have two terms with the variable p: -p and 35p. We also have a constant term: 35.

To combine like terms, we simply add or subtract their coefficients. The coefficient of -p is -1, and the coefficient of 35p is 35. So, we add these coefficients:

-1 + 35 = 34

Therefore, -p + 35p simplifies to 34p. Now, we can rewrite the expression as:

34p + 35

This step is crucial for simplifying the expression because it reduces the number of terms and makes the expression more concise. Combining like terms is a fundamental algebraic operation that simplifies expressions and makes them easier to understand and work with. By identifying and combining like terms, we are essentially grouping similar elements together, resulting in a more streamlined expression.

After applying the distributive property and combining like terms, the simplified form of the expression -p + 7(5 + 5p) is 34p + 35. This is the most simplified form of the expression, as there are no more like terms to combine and no parentheses to eliminate.

The final simplified expression consists of two terms:

  • 34p: This is a term with the variable p and a coefficient of 34. It represents 34 times the value of p.
  • 35: This is a constant term, a numerical value that does not change.

The simplified expression is much easier to understand and work with than the original expression. It clearly shows the relationship between the variable p and the constant term. This simplification process is crucial in algebra, as it allows us to solve equations, graph functions, and perform other mathematical operations more efficiently. The final simplified expression is the result of a systematic application of algebraic principles, demonstrating the power of simplification in making complex expressions more manageable.

In this article, we have successfully simplified the expression -p + 7(5 + 5p) to its simplest form, which is 34p + 35. We achieved this by following a step-by-step process:

  1. Applying the distributive property: We multiplied 7 by each term inside the parentheses (5 and 5p) to eliminate the parentheses.
  2. Combining like terms: We identified and combined the terms with the variable p (-p and 35p) to simplify the expression further.

This simplification process highlights the importance of understanding and applying algebraic principles. The distributive property and the ability to combine like terms are fundamental skills in algebra, enabling us to manipulate expressions and equations effectively. By mastering these skills, we can solve more complex mathematical problems and gain a deeper understanding of mathematical concepts.

Simplifying expressions is not just a mathematical exercise; it is a crucial skill that has applications in various fields, including science, engineering, and economics. The ability to break down complex problems into simpler components is essential for problem-solving in general. Therefore, understanding the simplification process is a valuable asset in both academic and real-world contexts.

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