Problem 68 - Algebra A Combined Function

Question

Use the distributive property to write each sum as a product. See Examples 13 and 14. $$ (-3) a+(-3) y $$

Step-by-Step Solution

Answer

The factored expression is \(-3(a + y)\).

Step 1: Identify the Common Factor

Look at the terms given in the sum: \(-3a\) and \(-3y\). Notice that \(-3\) appears in both terms as a factor.

Step 2: Apply the Distributive Property

The distributive property states that \(a(b+c) = ab + ac\). We can factor \(-3\) out of both terms. Write the expression as \(-3(a + y)\).

Step 3: Verify the Factored Expression

Expand \(-3(a + y)\) back using the distributive property to check: \(-3 \times a + (-3) \times y = -3a + (-3y)\). Since this equals the original expression, our factorization is correct.

Key Concepts

Factoring

In algebra, factoring is the process of breaking down an expression into a product of simpler expressions, or factors, that can be multiplied together to obtain the original expression. Factoring might seem complex, but it is an essential skill in solving algebraic equations and simplifying expressions. Think of it as "unwrapping" the expression into constituent parts. When applying factoring to algebraic expressions, you often look for the greatest common factor that each term in the expression shares. Once identified, the common factor is usually expressed outside of a set of parentheses, with the remaining simplified expression inside. This is akin to reversing the distributive property. Example: For the expression \(-3a + (-3y)\), the greatest common factor is \(-3\). Therefore, we factor it as \(-3(a + y)\). This process both simplifies the expression and lays the groundwork for solving more complex equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are used to represent quantities and relationships in algebra.These expressions can be as simple as a single number, like 7, or more complex, involving several terms connected by addition or subtraction. Each term in an expression is a product of numbers and variables raised to various powers.In the expression \(-3a + (-3y)\), you see both numbers and variables combined in a meaningful way.Components of Algebraic Expressions:

• Terms: The individual parts of an expression, separated by + or - signs. In our example, they are \(-3a\) and \(-3y\).
• Coefficients: The numerical part of terms, like the \(-3\) in both terms.
• Variables: Symbols used to represent unknown values or quantities, like letters \(a\) and \(y\).

Understanding algebraic expressions is key for manipulating them, particularly when applying properties like the distributive property for solving factorization problems.

Common Factor

The common factor is a number or expression that divides two or more terms without leaving a remainder. To identify a common factor, you look at each term in the expression and find any repeated number or variable.The process of finding common factors is crucial for simplifying expressions and solving equations efficiently. In our case, discovering that \(-3\) is a common factor of both \(-3a\) and \(-3y\) allows us to factor the expression as \(-3(a + y)\).Steps to Find Common Factors:

• Inspect each term in the expression.
• Identify numbers or variables that appear in every term.
• Use the greatest shared number or variable to "factor out" and simplify.

Identifying common factors not only helps in algebraic manipulation but also enhances understanding of how expressions can relate and simplify one another.