Problem 7 - Algebra for College Students

Question

State the property that justifies each of the statements. For example, \(3+(-4)=(-4)+3\) because of the commutative property of addition. $$-1(x+y)=-(x+y)$$

Step-by-Step Solution

Answer

This is the identity property of multiplication with respect to -1.

Step 1: Identify the Components

In the expression \(-1(x+y)=-(x+y)\), we are looking to identify if there's a mathematical property or rule being applied. Notice that \(-1\) is being distributed across the terms inside the parentheses \((x+y)\).

Step 2: Use the Identity Property

Consider the expression \(-1(x+y)\). The multiplication by \(-1\) on any expression will reverse its sign. The statement shows that \(-1(x+y)\) results in \(-(x+y)\), a direct application of the identity property of multiplication concerning \(-1\).

Key Concepts

Commutative Property

The commutative property is one of the foundational principles in algebra. This property refers to the ability to change the order of numbers involved in an operation without changing the result. It applies to both addition and multiplication. For addition, this means that \(a + b = b + a\). Likewise, for multiplication, it means that \(a \times b = b \times a\).

This property is particularly useful because it allows for flexibility in simplifying expressions and in operations involving large sets of numbers. It tells us that the order in which we add or multiply numbers does not affect the outcome, making computations more straightforward.

• Example for Addition: \(2 + 3 = 3 + 2\)
• Example for Multiplication: \(4 \times 5 = 5 \times 4\)

Overall, the commutative property helps in recognizing that certain sequences of operations can be rearranged for ease of calculation or understanding.

Identity Property

The identity property consists of the identity property of addition and the identity property of multiplication. These properties involve numbers that, when used in operations with other numbers, do not change the other number.

The addition identity property states that adding zero to any number does not change the value of that number. In algebraic terms, this can be demonstrated as \(a + 0 = a\). The number zero is considered the 'additive identity' because it keeps the identity of numbers when used in addition.

• Example: \(7 + 0 = 7\)

The multiplication identity property means multiplying any number by one does not change the identity of that number. It can be expressed as \(a \times 1 = a\). Here, one is the 'multiplicative identity'.

• Example: \(9 \times 1 = 9\)

In the provided exercise, the multiplication of \(-1(x+y)\) shows that multiplying by \(-1\) reverses the sign of the expression, illustrating an adaptation of the identity property in terms of negation: \(-1\) does not change the form (just the sign) of \((x+y)\).

Distributive Property

The distributive property is an essential tool in algebra for expressing and simplifying expressions. It connects multiplication and addition, allowing you to multiply a number by a sum by distributing the multiplication to each addend individually. It can be stated as \(a(b + c) = ab + ac\).

This property is particularly useful when simplifying algebraic expressions and solving equations. It helps in breaking down complex expressions into manageable parts and is regularly used to simplify equations.

• Example: Simplifying \(3(2 + 4)\) can be done by distributing 3 across both terms inside the parenthesis: \(3 \times 2 + 3 \times 4 = 6 + 12 = 18\).

In the given exercise, the expression \(-1(x+y)\) showcases the use of the distributive property, where \(-1\) is distributed across each term inside the parenthesis, leading to \(-(x+y)\). This illustrates how multiplication operates across additions inside parentheses, applying consistent sign changes.