Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ \frac{x+y}{x+1}-\frac{y}{x}=\frac{x^{2}-y}{x(x+1)} $$

Step-by-Step Solution

Answer

The real number property that justifies the statement \( \frac{x+y}{x+1} - \frac{y}{x} = \frac{x^2-y}{x(x+1)} \) is the closure property for addition, subtraction, and multiplication. This property states that the sum, difference, or product of two real numbers is also a real number. By finding a common denominator and using proper fraction properties, we showed that the original statement is true.

Step 1: Identify the common denominator

We start by finding the common denominator between the fractions. In this case, the denominators are x and (x+1). The common denominator will be their product, which is x(x+1).

Step 2: Rewrite the expression using the common denominator

Now, we'll rewrite each term of the left side of the equation using the common denominator we found in Step 1: \( \frac{x+y}{x+1} - \frac{y}{x} \) = \( \frac{(x+y)(x)}{x(x+1)} - \frac{y(x+1)}{x(x+1)} \)

Step 3: Perform operations in the numerator

Once we have the common denominator, we can now perform the operations within the numerators to simplify the expression: \(=\frac{(x^2+xy)}{x(x+1)} - \frac{(yx+y)}{x(x+1)}\)

Step 4: Combine terms with the same denominator

All terms on the left side of the equation have the same denominator, so we can combine these terms: \(=\frac{(x^2+xy)-(yx+y)}{x(x+1)}\)

Step 5: Simplify remaining expression

Now we can simplify the expression by cancelling out common terms in the numerator: \(=\frac{x^2+xy-yx-y}{x(x+1)}\) \(=\frac{x^2-y}{x(x+1)}\)

Step 6: Conclusion

Now, comparing the final expression to the original statement, they are both the same: \( \frac{x+y}{x+1} - \frac{y}{x} = \frac{x^2-y}{x(x+1)} \) This solution is justified by the real number property called "the closure property" for addition, subtraction, and multiplication, which states that the sum, difference, or product of two real numbers is also a real number. By finding a common denominator and using proper fraction properties, we showed that the original statement is true.

Key Concepts

Common Denominator

Understanding the concept of a common denominator is integral to working with fractions in mathematics. It refers to a shared multiple of the denominators of two or more fractions. When adding, subtracting, or comparing fractions, a common denominator allows for straightforward calculations by creating a uniform base.

Here's a simplified approach:

• Identify the denominators in the given fractions.
• Determine if one denominator is a multiple of the other(s), or if they are relatively prime (share no common factors other than 1).
• If there's no common multiple, multiply the denominators together to obtain the common denominator.
• Rewrite each fraction so that their denominators match this common base.
• Adjust their numerators accordingly to reflect this change in base.

Remember, the objective is not to find the smallest common denominator but any common denominator that enables us to simplify the expression.

Simplifying Expressions

The process of simplifying expressions is a fundamental step in solving algebraic problems. It involves reducing expressions to their simplest form, making them more understandable and easier to work with. This may include combining like terms, factoring, canceling common factors, and simplifying complex fractions.

Here's how to simplify effectively:

• Look for and combine like terms, which are terms that have the same variable raised to the same power.
• Factor expressions to reveal common factors that can be cancelled out.
• Apply the distributive property where required to eliminate parentheses.
• Remember to perform all operations according to the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Simplifying not only makes the expressions more manageable but also prepares them for further operations, such as solving equations or evaluating variables.

Algebraic Fractions

Working with algebraic fractions can sometimes be intimidating, but they follow the same principles as arithmetic fractions. An algebraic fraction has a numerator and denominator that are algebraic expressions, often containing variables.

To manipulate these fractions effectively, remember these steps:

• When combining algebraic fractions, first find a common denominator.
• Rewrite each fraction using this common denominator.
• This will often allow you to combine the fractions, simplify complex numerators, or cancel out factors.
• Always look to factor where possible, as this can simplify the fraction further.

In the context of the exercise given, finding a common denominator allowed for the algebraic fractions to be combined and simplified. By understanding and applying these principles, algebraic fractions become less complex and more approachable.