Problem 83 - Algebra 2

Question

PREREQUISITE SKILL Identify the additive inverse for each number or expression. \(5-6 y\)

Step-by-Step Solution

Answer

The additive inverse of \(5 - 6y\) is \(-5 + 6y\).

Step 1: Understand Additive Inverse

The additive inverse of a number or expression is what you add to it to get zero. For any number or expression \( x \), its additive inverse is \( -x \). The two numbers sum to zero.

Step 2: Apply to the Expression

We start with the expression \( 5 - 6y \). To find the additive inverse of this expression, we will change the signs of each term in the expression.

Step 3: Change the Signs

The expression \( 5 - 6y \) has two terms: \( 5 \) and \( -6y \). The additive inverse of \( 5 \) is \( -5 \), and the additive inverse of \(-6y \) is \( 6y \).

Step 4: Combine Inverses

Combine the additive inverses of each term to get the additive inverse of the expression: \(-5 + 6y \).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. These expressions do not have an equal sign, unlike equations.
They might look like simple or complex sentences in mathematics, depending on the number of terms involved. For instance, in the expression \(5 - 6y\), "5" is a constant, and "\(-6y\)" is a term that includes the variable "\(y\)".
Variables like "\(y\)" represent unknown values that can change, and they make algebraic expressions dynamic. To understand such expressions, one needs to know how to manage constants and variables together, using operations appropriately.

Inverse Operations

Inverse operations are mathematical methods used to reverse or undo the effects of another operation.
The concept of inverse operations is crucial in solving equations and manipulating algebraic expressions, as they help us find unknowns by reversing actions.
In the context of additive inverse, the operation involves finding a number that, when added to the original number, results in zero.

• For addition, the inverse operation is subtraction and vice versa.
• For multiplication, the inverse is division and vice versa.

Specifically, the additive inverse of an expression like \(5 - 6y\) involves changing the signs of the terms to become \(-5 + 6y\). This operation effectively "undoes" the original expression's numerical effect, leading to a net value of zero when combined with the original.

Prerequisite Skills

Before diving into finding additive inverses, it's essential to have a strong grasp of some foundational skills.
Understanding these prerequisite skills helps students approach algebraic problems with confidence and ease.

• **Basic Operations**: Mastery in addition, subtraction, multiplication, and division is crucial as these are the backbone of algebraic manipulations.
• **Comprehending Variables**: Recognizing that variables stand for unknown values and learning how to handle them is fundamental to working with algebraic expressions.
• **Sign-Changing**: Knowing how to change the signs of numbers and terms is key to identifying additive inverses and balancing equations.

This foundation supports the development of more complex algebraic thinking, setting a solid base for tackling a wide array of mathematical tasks.