Problem 61 - Algebra 1

Question

Formulate the following statement in terms of variables. Then decide whether it is true or false . The opposite of the sum of two numbers is equal to the sum of the opposites of the numbers. If false, give a counterexample. If true, give two examples involving negative numbers.

Step-by-Step Solution

Answer

The statement is true; '-(a + b)' equals '-a - b'. Two examples with some negative numbers are a=-2, b=-3, and a=-1, b=-4.

Step 1: Formulate the statement in terms of variables

Let's use 'a' and 'b' as the two numbers. According to the given statement, the opposite of the sum of two numbers is '-(a + b)'. This should be equivalent to the sum of the opposites of the numbers ' -a - b'.

Step 2: Evaluate if the statement is true

To evaluate if the given statement is true, it is necessary to check whether '-(a + b)' equals '-a - b'. This is legitimized through the mathematical property known as 'Distributive Property of Multiplication over Addition', which suggests that '-1 * (a+b)' is equal to '-1*a + -1*b'. So the two expressions are equivalent, meaning the statement is true.

Step 3: Give examples

Two examples involving negative numbers could be: \nExample 1: Let 'a = -2' and 'b = -3'. Then, '-((-2) + (-3)) = -(-5) = 5' which is equal to ' -(-2) -(-3) = 2 + 3 = 5'.\n Example 2: Let 'a = -1' and 'b = -4'. Then, '-((-1) + (-4)) = -(-5) = 5' which is equal to ' -(-1) - (-4) = 1 + 4 = 5'.

Key Concepts

Variables and Their Importance in Algebra

Variables represent unspecified numbers in algebra, symbolized commonly using letters like "a" or "b". They are quite handy when formulating general mathematical statements or equations. For instance, saying "the opposite of the sum of two numbers" might sound abstract. However, using variables, we rephrase it as "the opposite of \((a + b)\)."
So, variables allow us to write universal statements applicable to all numbers. They convert verbal statements into a mathematical form that we can test or manipulate, which is essential when solving algebraic problems.
Variables provide a way to express relationships between numbers, making them a fundamental aspect of algebra. In the context of the problem, variables 'a' and 'b' helped to encapsulate the sum and its opposite, making it easier to check mathematical properties like the Distributive Property.

Understanding Negative Numbers

Negative numbers are numbers less than zero and are represented with a minus sign (-). Understanding these numbers is crucial in mathematics, especially when working on equations or inequalities.
When you encounter expressions like "the opposite of the sum," it involves negating a number, essentially multiplying it by -1. For example, if the sum \(a+b\) equals -5, the opposite would be \(-(-5) = 5\).

• Negative numbers behave in interesting ways during operations: when you add two negative numbers, the result is more negative.
• When you multiply two negative numbers, however, the result is positive, due to the negatives canceling each other out.
• Understanding how these numbers interact within expressions is key to solving many mathematical problems.

This foundational knowledge about negative numbers facilitates explorations into their properties and operations, ensuring you correctly apply mathematical principles like those involving variables and sums.

Core Mathematical Properties

Mathematical properties are rules and laws that form the basis upon which mathematics works. The Distributive Property is one of the essential properties, particularly when dealing with expressions involving variables and negative numbers.
The Distributive Property of Multiplication over Addition states that \(-1 \cdot (a + b)\) is the same as \(-1 \cdot a + -1 \cdot b\). It means distributing the -1 to both terms inside the parentheses, resulting in \(-a - b\). This property is vital when verifying algebraic statements for their truth value.

• This property simplifies complex problems by breaking them into more manageable parts, making evaluation simpler.
• In our given example, it helped establish the equivalence of two expressions \(-(a+b)\) and \(-a - b\).
• Applying the Distributive Property wisely ensures accurate computation, avoiding errors in solving equations.

Familiarity with these properties not only aids in solving problems but also strengthens your understanding of mathematics, allowing more advanced exploration of analytical pursuits.