Problem 65 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each sum as a product. See Example 5 \(11 x+11 y\)

Step-by-Step Solution

Answer

The sum \(11x + 11y\) can be written as the product \(11(x + y)\).

Step 1: Identify the Common Factor

Look at the expression \(11x + 11y\). Notice that both terms share a common factor, which is \(11\).

Step 2: Apply the Distributive Property

Use the distributive property, which states that \(a(b + c) = ab + ac\). Here, factor out the common factor from both terms: \(11(x + y)\).

Step 3: Verify the Factorization

To verify, distribute \(11\) back into the expression: \(11 \times x = 11x\) and \(11 \times y = 11y\). This gives the original sum, confirming the factorization is correct.

Key Concepts

Factoring

Factoring is a fundamental concept in algebra that involves expressing an expression as a product of its factors. Essentially, it is like breaking down numbers or expressions into simpler multiplicative components. In the case of algebraic expressions, factoring can make complex problems easier to solve or simplify.For example, when you have an expression like \(11x + 11y\), factoring involves finding a way to express this sum as a product of terms. This can help in simplifying expressions or solving equations where setting expressions to zero can help us find roots. In our case, we identified that \(11\) is a common factor, which allowed us to express the original expression as \(11(x + y)\).
Factoring can reveal important properties of algebraic expressions and is often the first step in solving algebraic equations, simplifying fractions, and more. If you practice finding and using factors, you'll gain a better understanding of how algebraic structures work.

Common Factor

A common factor is an element that appears in each term of an expression. It is one of the key components when applying the distributive property to factor expressions. Identifying common factors is a crucial step in simplifying algebraic expressions.In the expression \(11x + 11y\), the number \(11\) is a common factor because it divides evenly into both terms. Finding it involves looking at what's similar across different terms of an expression.

• Identify the coefficient(s) in each term.
• Look for the greatest common number in those coefficients.
• The highest number that all terms can be divided by is your common factor.

Once identified, the common factor can be factored out, simplifying the overall expression. This makes it easier to manipulate the expression in future calculations or solve equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They serve as a building block for algebraic equations and are a representation of mathematical relationships.For instance, \(11x + 11y\) is an algebraic expression consisting of two terms. Here, \(x\) and \(y\) are variables while \(11\) is a coefficient. Expressions can be simple or complex, combining multiple terms through addition, subtraction, multiplication, or division.

• Terms: Parts of an expression separated by + or - signs.
• Coefficients: Numbers multiplying the variables.
• Variables: Symbols representing unknown or variable numbers.

Understanding how to manipulate algebraic expressions, like factoring or expanding them, is key to solving algebraic equations. Algebraic expressions are the language of algebra, and mastering them will aid in tackling more complex mathematical challenges.