Problem 3 - 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. $$42+(-17)=-17+42$$

Step-by-Step Solution

Answer

Commutative property of addition.

Step 1: Identify the Operation

Examine the given expression: \(42 + (-17) = -17 + 42\). Notice that both sides of the equation involve addition.

Step 2: Recognize the Property

Recall the properties of addition. The commutative property of addition states that the order of numbers being added can change without affecting the sum: \(a + b = b + a\).

Step 3: Apply the Property

Apply the commutative property of addition to the statement. The left-hand side \(42 + (-17)\) becomes the right-hand side \(-17 + 42\) by changing the order of the numbers.

Step 4: State the Property

Confirm and state that the expression \(42 + (-17) = -17 + 42\) is justified by the commutative property of addition, as it shows the order of the addends is switched.

Key Concepts

Addition Properties

Understanding addition properties can help make math easier. One of the key properties is the **commutative property**. It allows us to change the order of numbers that we are adding together. This property can be written as: \(a + b = b + a\). It means that no matter how you arrange the numbers, the sum remains the same. For example, if you have 3 + 5, it is the same as 5 + 3. Both expressions equal 8.

• **Commutative Property:** The order of addends doesn't change the sum.
• **Associative Property:** Grouping of addends doesn't change the sum.
• **Identity Property:** Adding zero keeps the same number.

Being familiar with these properties can reduce mistakes and make calculations quicker. Plus, when solving equations, recognizing these properties helps to simplify and solve them much more effectively.

Algebra

Algebra is a branch of mathematics that uses symbols or letters to represent numbers and express mathematical ideas. In algebra, properties like the commutative property are extremely useful.

• We often use **variables** (like \(x\) or \(y\)) to represent unknown values.
• **Algebraic expressions** can include numbers, variables, and operators like plus and minus.
• Algebra is fundamental in solving equations and finding unknowns.

Knowing algebra allows you to compose, comprehend, and solve various mathematical problems. For example, if \(x + 2 = 5\), using subtraction, we find what number \(x\) must be to make the equation true. Here, understanding algebra lets you rearrange and manipulate equations to identify unknowns.

Mathematical Reasoning

Mathematical reasoning involves using logical thinking to connect ideas and solve problems. It is the process of applying mathematics in reasoning scenarios.

• It includes recognizing patterns, forming conjectures, and proving them.
• Logical steps are crucial in justifying every solution made.
• Helps build confidence in problem-solving by verifying each step.

For example, if we want to show that \(42 + (-17) = -17 + 42\), mathematical reasoning through the commutative property of addition helps us confirm this equality. When we use mathematical reasoning, we look at what we know (the properties) and what we need to find out, to reason our way to the solution.