Problem 47 - Algebra 2

Question

NUMBER THEORY For Exercises \(46-49,\) use the properties of real numbers to answer each question. If \(m+n=0,\) what is the value of \(n ?\) What is \(n^{\prime}\) s relationship to \(m ?\)

Step-by-Step Solution

Answer

The value of \( n \) is \( -m \), making \( n \) the additive inverse of \( m \).

Step 1: Understand the Given Equation

The exercise provides us with the equation \( m + n = 0 \). Our task is to determine the value of \( n \) from this equation and understand its relationship to \( m \).

Step 2: Solve the Equation for n

To find the value of \( n \), we rearrange the equation. Start with the given equation: \( m + n = 0 \). Subtract \( m \) from both sides to isolate \( n \): \( n = -m \).

Step 3: Determine the Relationship Between n and m

From the equation \( n = -m \), we see that \( n \) is the additive inverse of \( m \). This means that when \( m \) is added to \( n \), they sum to zero, which is consistent with the property that a number plus its additive inverse equals zero.

Key Concepts

Properties of Real Numbers

The properties of real numbers are fundamental rules that apply to all real numbers. This includes any number you can think of: positive, negative, fractional, or whole numbers. Understanding these properties can greatly simplify the process of solving equations. Some important properties are:

• Commutative Property: The order of addition or multiplication does not change the result. For example, \(a + b = b + a\) and \(a \times b = b \times a\).
• Associative Property: The way numbers are grouped in addition or multiplication does not change their sum or product. For example, \((a + b) + c = a + (b + c)\) and \((a \times b) \times c = a \times (b \times c)\).
• Distributive Property: This property links addition and multiplication, such as \(a(b + c) = ab + ac\).
• Identity Property: Adding zero or multiplying by one leaves the original number unchanged. Thus, \(a + 0 = a\) and \(a \times 1 = a\).

In the context of the problem, the most relevant property is the implication that when two numbers add up to zero, each is the additive inverse of the other.

Additive Inverse

The additive inverse of a number is what you add to that number to get zero. This idea plays a crucial role in our problem-solving exercise. If you have a number \(m\), then its additive inverse is \(-m\) because \(m + (-m) = 0\).
In our exercise, given that \(m + n = 0\), it follows that \(n = -m\). This clearly shows us that \(n\) and \(m\) are additive inverses. Their sum remains zero because anytime a number is added to its additive inverse, their effects cancel out completely.
Understanding this concept helps simplify many algebraic processes, especially in equation solving where we frequently need to find an unknown number that pairs with another to result in zero.

Equation Solving

Equation solving is a vital skill in mathematics. It involves finding the value of unknown variables that satisfy a particular equation. Let's review the basic steps to solve equations, using the exercise equation \(m + n = 0\) as an example:

• Identify the Equation: Recognize the equation you're working with, in this case, \(m + n = 0\).
• Isolate the Variable: Rearrange the equation to get the unknown variable alone. Here, we subtract \(m\) from both sides to get \(n = -m\).
• Verify Your Solution: Double-check the solution by plugging it back into the original equation to confirm it holds true. Substitute \(n\) with \(-m\): \(m + (-m) = 0\), which indeed results in zero.

This problem involves a linear equation, which is among the simplest forms to solve. However, mastering the basic problem-solving process is crucial, as it lays the foundation for tackling more complex mathematical problems.