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Chapter 1: Problem 806

Simplify using the distributive property. $$ 8-(r-7) $$

Short Answer

Expert verified
15 - r

Step by step solution

01

Step 1- Identify the Distributive Property

The distributive property in algebra states that for any numbers or variables, the expression typically in the form of a(b - c) or a(b - c) can be rewritten as a multiplied by each term within the parentheses. In this problem, the expression given is 8-(r-7).
02

- Distribute any Negative Signs

Notice that there is a negative sign right before the expression r-7. We need to distribute this negative sign through the expression inside the parenthesis. This means that - (-1) will multiply r and -7 : 8 - (r) + (-(-7) = 8 - r +7.
03

- Simplify the Expression

Combine the constant terms (8 and 7 together): 8 + 7 - r, which simplifies to 15 - r.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebra Simplification
Algebra simplification is the process of making algebraic expressions more manageable and straightforward. By simplifying expressions, we make them easier to understand and solve.
The main strategies to simplify algebraic expressions include:
  • Distributing terms
  • Breaking down expressions
  • Combining like terms
  • Removing unnecessary parentheses
In the given exercise, we begin by identifying the distributive property to handle the terms inside the parentheses. This process helps to break down and simplify the initial complex expression into a more straightforward form.
Negative Signs Distribution
When dealing with expressions in algebra, distributing negative signs correctly is essential.
In our exercise, we see that the expression is written as:
\[ 8 - (r - 7) \]
We need to distribute the negative sign (-) across the terms inside the parentheses. Think of the negative sign as multiplying each term by -1:
\[ -1 * (r - 7) \]
Distributing the negative sign, we get:
\[ -(r) + -(-7) = -r + 7 \]
So our original expression now looks like this:
\[ 8 - r + 7 \]
This careful handling of negative signs is crucial for accuracy in algebra.
Combining Like Terms
Combining like terms is an essential part of simplifying algebraic expressions. It involves summing up coefficients of similar terms to reduce the expression further.
Like terms are terms that have the same variable raised to the same power. In our simplified expression from the exercise:
\[ 8 - r + 7 \]
We see that 8 and 7 are like terms because they are both constants. By combining these like terms, we get:
\[ 8 + 7 = 15 \]
Therefore, the expression is now:
\[ 15 - r \]
This final simplified form is easier to handle and more intuitive. Combining like terms is a fundamental skill that helps in solving equations and understanding their solutions better.

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