Problem 2 - Algebra 1: Concepts and Skills

Question

Match the property of equality with its description. Addition Property of Equality A. If \(a=b,\) then \(c a=c b\) B. If \(a=b,\) then \(a-c=b-c\) C. If \(a=b,\) then \(a+c=b+c\) D. If \(a=b\) and \(c \neq 0,\) then \(\frac{a}{c}=\frac{b}{c}\)

Step-by-Step Solution

Answer

The Addition Property of Equality matches with Option C.

Step 1: Understanding Addition Property of Equality

The Addition Property of Equality states that: If the same amount is added to both sides of an equation, then the equality still holds true.

Step 2: Match the property with the description

Now we need to find which of the given options matches up with the Addition Property of Equality. Looking at the options, option C states: If \(a=b,\) then \(a+c=b+c\), which is adding the same amount to both sides of an equation, which is precisely the definition of the Addition Property of Equality.

Step 3: Verify the Solution

Through the process of elimination, verify that no other option correctly represents the Addition Property of Equality. Options A, B and D involve multiplication, subtraction or division, none of them represent the Addition Property of Equality.

Key Concepts

Equation Solving

Equation solving is like the heart of basic algebra. It is all about finding the value of the variable that makes the equation true. An equation is a mathematical statement that asserts the equality of two expressions.
For instance, if you have an equation like \( x + 5 = 12 \), solving it means determining what value of \( x \) will make both sides equal. Here, you'd subtract 5 from both sides to find that \( x = 7 \).

Key steps in solving equations include:

• Identifying the operations needed (like addition, subtraction, multiplication, or division).
• Applying the right properties of equality to maintain the equation balance.
• Systematically working to isolate the variable.

Remember, the main goal is to simplify the equation and accurately solve for the unknown variable.

Equality Properties

Equality properties are the rules in mathematics that allow us to manipulate equations while keeping them balanced. They provide the foundation for solving equations reliably and logically.
Here are some of the main properties:

• Addition Property of Equality: If \(a = b\), then \(a + c = b + c\). You can add the same number to both sides without changing the equation's validity.
• Subtraction Property of Equality: If \(a = b\), then \(a - c = b - c\). Subtraction can maintain equality similarly to addition.
• Multiplication Property of Equality: If \(a = b\), then \(ca = cb\). Multiplying both sides by the same number doesn't affect the equality.
• Division Property of Equality: If \(a = b\) and \(c eq 0\), then \(\frac{a}{c} = \frac{b}{c}\). Divide by a non-zero number, and it still holds true.

Understanding these properties helps in aligning the equation solving steps in the right direction.

Mathematical Reasoning

Mathematical reasoning involves using logic to solve problems and prove assertions in mathematics. It requires critical thinking and an understanding of mathematical principles.
Here's how you can think about it:

• Identify the known values and what you need to find.
• Use properties of equality to justify each step in solving equations.
• Employ strategies like breaking down complex problems into simpler parts.

Effective mathematical reasoning allows you to validate your solutions and ensures that you have followed a logical pathway to arrive at the solution. It involves ensuring each step in a mathematical process is sound, making the use of properties like those for equality straightforward and effective.