Problem 35 - Algebra 1

Question

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -r(r-9) $$

Step-by-Step Solution

Answer

The expression \(-r(r-9)\) without parentheses is \(-r^2 + 9r\).

Step 1: Distribute -r to each term inside the parentheses

In order to rewrite the expression without parentheses, distribute \(-r\) to each of the terms inside the parenthesis: \[ -r \cdot r + -r \cdot -9)\]

Step 2: Multiply

Now, perform the multiplication operation for both terms: \[ -r^2 + 9r \]

Step 3: Simplify if possible

In this case, the equation is already in its simplest form, so there is no further simplification available. The rewritten expression without parentheses is thus:\[ -r^2 + 9r \]

Key Concepts

Algebraic Expressions

Algebraic expressions are an essential part of mathematics. They consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Unlike equations, expressions don't have an equals sign. They represent a value that can change depending on the values of the variables involved.

In the expression \(-r(r-9)\), the variable is "\(r\).” Variables can stand for any number, which means the value of an algebraic expression can change with different inputs. Understanding how to manipulate these expressions helps in solving problems effectively.

• Variables: Letters like \(r\) that can represent any number.
• Constants: Fixed numbers in the expression like 9.
• Operations: Math actions like addition and subtraction used between variables and numbers.

Grasping the structure of algebraic expressions enables you to simplify, solve, and manipulate them with greater ease, crucial in both basic and advanced math topics.

Simplification

Simplification in algebra involves making an expression easier to read or work with by combining like terms or performing calculations. It is a key skill in problem-solving as it transforms complex expressions into more manageable forms.

To simplify, we often follow these steps:

• Distribute any multiplications over additions or subtractions.
• Combine like terms, which are terms with the same variables raised to the same power.
• Perform any calculations or operations as needed.

In our exercise, after applying the distributive property, the expression \(-r^2 + 9r\) is already simplified. Each term is distinct, meaning there are no like terms to combine. This complete simplification can help in further steps of problem-solving or in setting up equations.

By simplifying an expression, you reduce likely errors and make further calculations simpler, supporting more advanced algebraic work.

Parentheses in Algebra

Parentheses are crucial in algebra to indicate which operations should be performed first and how expressions are grouped. They help in ensuring the intended operations are executed correctly, especially when multiple operations are involved.

In the original exercise, parentheses are used to show that the subtraction inside them, \((r-9)\), must be considered before multiplication by \(-r\). The distributive property, which is heavily used in algebra, allows you to "break open" the parentheses and spread the multiplication over each term inside.

This is accomplished through the distributive law of multiplication, which states that\[(a(b+c) = ab + ac)\]. The order of combining or distributing terms is vital to preserving the mathematical correctness of the expression. After distribution, the expression \(-r(r-9)\) turns into \(-r^2 + 9r\), fully removing the parentheses while keeping the mathematical integrity of the expression.

• Ensure correct order by performing operations inside parentheses first.
• Use the distributive property to remove parentheses systematically.
• Check that all operations align with initial order of operations before solving further.

Using parentheses correctly guides accurate problem-solving and prevents unnecessary mistakes, forming a foundation of sound algebra practice.