Problem 66 - Algebra A Combined Function

Question

Find each multiplicative inverse or reciprocal. $$ 5 $$

Step-by-Step Solution

Answer

The multiplicative inverse of 5 is \( \frac{1}{5} \).

Step 1: Understanding the Multiplicative Inverse

The multiplicative inverse of a number is another number which, when multiplied with the original number, results in 1. This means for a number \( a \), its multiplicative inverse \( a^{-1} \) satisfies the equation \( a \times a^{-1} = 1 \).

Step 2: Finding the Multiplicative Inverse of 5

For the number 5, we need to find a number \( x \) such that \( 5 \times x = 1 \). To isolate \( x \), divide both sides of the equation by 5. Hence, \( x = \frac{1}{5} \).

Step 3: Verification

Let's verify our answer by substituting back into the equation. Multiply 5 by \( \frac{1}{5} \) to check if the product is indeed 1: \( 5 \times \frac{1}{5} = 1 \). This verifies that the multiplicative inverse of 5 is correct.

Key Concepts

Reciprocal

In mathematics, the reciprocal of a number is simply another term used to describe its multiplicative inverse. To find the reciprocal of any non-zero number, you need to divide 1 by that number. For example, the reciprocal of 5 is \( \frac{1}{5} \).
The action of finding a reciprocal involves flipping the fraction form of the number. Therefore, if a number is expressed as a fraction like \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \).

• Reciprocal of whole number 5 equals \( \frac{1}{5} \)
• Reciprocal of a fraction \( \frac{3}{4} \) is \( \frac{4}{3} \)

It's essential to remember that zero does not have a reciprocal because no number multiplied by zero equals one.

Algebraic Concepts

Algebraic concepts form the basis for solving equations and understanding mathematical relationships. The multiplicative inverse is a key algebraic concept that helps balance equations and solve for variables.
In algebra, when you want to find the reciprocal or multiplicative inverse of a number, especially in solving equations, you use this concept to "undo" multiplication, effectively isolating the variable or achieving a simpler form of expression.

• By applying the multiplicative inverse, you can solve equations like \( ax = 1 \) by dividing both sides by \( a \).
• This concept helps understand that operations can be reversed, which is fundamental when working with equations.

Understanding algebraic concepts like this one helps in solving complex problems by simplifying them into smaller, manageable tasks.

Equations

Equations in mathematics express the equality between two expressions, often involving variables and constants. Solving equations requires an understanding of various methods and concepts, such as reciprocal and multiplicative inverse.
When dealing with simple equations like \( 5x = 1 \), finding the value of \( x \) involves determining the multiplicative inverse of the number multiplying \( x \). In this case, divide both sides by 5, resulting in \( x = \frac{1}{5} \).

• Equations can be linear, quadratic, or more complex, requiring different strategies for solving.
• The goal is always to isolate the variable and find its value.

Equations are more than simple calculations; they represent relationships between variables and are used extensively in real-life scenarios, from physics to economics.