Problem 6 - Elementary Algebra

Question

For Problems \(1-10\), find the indicated products by applying the distributive property; for example, $$ \begin{aligned} (x+1)(y+5) &=x(y)+x(5)+1(y)+1(5) \\ &=x y+5 x+y+5 \end{aligned} $$ $$ (x-7)(y-9) $$

Step-by-Step Solution

Answer

The product is \(xy - 9x - 7y + 63\).

Step 1: Identify Terms of Each Binomial

The first binomial is \((x - 7)\), and it has terms \(x\) and \(-7\). The second binomial is \((y - 9)\), and it has terms \(y\) and \(-9\).

Step 2: Apply the Distributive Property

Distribute each term of the first binomial to each term of the second binomial. This involves multiplying each term in \((x - 7)\) by each term in \((y - 9)\).

Step 3: Multiply Each Pair of Terms

Perform the following multiplications and combine the results:- \(x \times y = xy\)- \(x \times (-9) = -9x\)- \(-7 \times y = -7y\)- \(-7 \times (-9) = 63\)

Step 4: Combine All Results

Combine all the terms obtained from the multiplication to form the expression: \(xy - 9x - 7y + 63\).

Key Concepts

Understanding Binomials

A binomial is a type of polynomial that consists of exactly two terms. These terms are usually joined by a plus or minus sign. Because of their simple structure, they form an essential building block for more complex algebraic expressions. The key components of a binomial include coefficients and variables. For example, in the expression \(x + 1\), \(x\) and \(1\) are the two terms.
Binomials are common in algebra and often appear in problems involving polynomial multiplication. When multiplying binomials, it is crucial to consider each term individually, as each must be multiplied with every term in the other binomial. This process ensures that all possible products are accounted for, which is foundational for polynomial multiplication.

Mastering Polynomial Multiplication

Polynomial multiplication involves finding the product of two or more polynomials. This can be a daunting task, but understanding the distributive property simplifies it. When multiplying two binomials like \((x-7)(y-9)\), we apply this property by multiplying each term in the first binomial by each term in the second binomial.
Let's break it down:

• Multiply \(x\) by \(y\) to get \(xy\).
• Multiply \(x\) by \(-9\) to get \(-9x\).
• Multiply \(-7\) by \(y\) to get \(-7y\).
• Multiply \(-7\) by \(-9\) to get \(63\).
  

Afterward, you sum up all these terms to get the final expression: \(xy - 9x - 7y + 63\). This systematic approach ensures every possible combination is covered, resulting in a comprehensive solution.

Exploring Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (such as addition and multiplication) that represents a particular value. These expressions are versatile and form the basis of algebra.
In the exercise (\(x-7\)(\(y-9\))), we work with algebraic expressions that entail variables \(x\) and \(y\). The aim is to simplify these expressions through multiplication and combination of like terms. Variables serve as placeholders for numbers, which means the expression can take on various values depending on the numbers substituted for \(x\) and \(y\).
Algebraic expressions are essential not only for solving equations but also for modeling real-world scenarios in mathematics. Understanding how to manipulate and evaluate these expressions is a critical skill in algebra.