Problem 6 - Algebra 2

Question

Name the property illustrated by each statement. If \(4 c=15,\) then \(4 c+2=15+2\)

Step-by-Step Solution

Answer

The property illustrated is the Addition Property of Equality.

Step 1: Identify Given Expression

Take a close look at the expression provided, which consists of two equalities: \(4c = 15\) and \(4c + 2 = 15 + 2\). Our goal is to identify what algebraic property is demonstrated by the transition between these two equations.

Step 2: Transitional Step Analysis

Notice the alteration made to both sides of the initial equation \(4c = 15\). Each side of the equation has been incremented by 2, suggesting that this step involves adding a constant to both sides.

Step 3: Determine the Property

Based on the addition of a constant to both sides of the equality, the property in question can be identified as the "Addition Property of Equality." This property states that if two expressions are equal, adding the same number to both sides maintains the equality.

Key Concepts

Addition Property of Equality

The Addition Property of Equality is a core algebraic principle that can help us solve equations smoothly. This property states that if we have two equal expressions, we can add the same value to both sides without changing the equality.

Imagine you have a balanced scale; if you add the exact same weight to both pans, the scale remains balanced. This is just like an equation where, adding a certain number to both sides keeps it true.

• For instance, in the exercise: If we start with the equation \(4c = 15\), and add 2 to both sides, it becomes \(4c + 2 = 15 + 2\).
• Notice how both sides of the equals sign changed in the same way, and that's why this equality holds.

Understanding this property is crucial for maintaining the balance needed to solve algebraic equations effectively.

Equality

Equality is a fundamental concept in algebra and mathematics as a whole. It means that two expressions are exactly the same in terms of value.

When we talk about equations, we're usually discussing equalities. An equation like \(4c = 15\) tells us that whatever value we put in place of \(c\), after multiplying by 4, should give us 15.

• It represents harmony and balance in math, as both sides of the equal sign must have the same value.
• If we change one side but not the other, the equation will be false unless we use a property like the addition property.

Grasping equality is vital, as it forms the basis of finding unknown values and solving problems mathematically.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition or subtraction) that collectively represent a quantity.

In our exercise, several expressions are present: \(4c\), \(15\), \(4c + 2\), and \(15 + 2\). We use these expressions to form equations and inequalities.

• Expressions such as \(4c\) mean "4 multiplied by \(c\)", where \(c\) is a variable.
• Understanding expressions is essential because they are the building blocks of all equations.
• Through manipulating expressions using different algebraic properties, like the addition property, we solve equations and uncover unknowns.

Familiarity with different expressions and how they work can significantly enhance your ability to tackle algebraic problems successfully.