Problem 52 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$2(y+9)$$

Step-by-Step Solution

Answer

Question: Expand the expression using the distributive property: $$2(y+9)$$ Answer: $$2y + 18$$

Step 1: Identify the terms to be distributed

In the given expression, $$2(y+9)$$, we will multiply 2 by both the terms inside the parentheses, i.e., y and 9.

Step 2: Apply the distributive property

Multiply 2 by y and 9 separately. So, we have: $$2\times y + 2\times 9$$

Step 3: Simplify the expression

Now, perform the multiplications and simplify the expression: $$2y + 18$$ The expanded expression using the distributive property is $$2y + 18$$.

Key Concepts

Understanding Algebra

Algebra acts as a cornerstone in the world of mathematics, serving as the foundation for numerous mathematical concepts. It deals with symbols and the rules for manipulating these symbols to solve problems. Algebraic expressions, like the one given in the exercise \(2(y+9)\), are composed of numbers, variables (like \(y\)), and arithmetic operations (such as addition and multiplication).

In algebra, it's essential to understand that expressions represent quantities that can vary, which is why we use variables. Variables allow us to describe general rules and relationships between quantities, and solving algebraic expressions often involves finding the value of these variables.

Expanding Expressions

The process of expanding expressions involves transforming a compact algebraic expression into an equivalent extended form. This is often done to express the equation in a more straightforward manner or to simplify the process of solving it. To expand an algebraic expression, you might use the distributive property, which is precisely what the original exercise demonstrates.

The distributive property states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). The property comes in very handy when you encounter an expression where a single term is multiplied by a sum or difference within parentheses. When performing the expansion, each term inside the parentheses gets multiplied by the term outside, helping to 'distribute' the multiplication over addition or subtraction, hence the name.

Simplifying Expressions

Simplifying expressions is a vital skill in algebra. This process involves altering the form of an expression without changing its value. The aim is to make the expression as straightforward as possible. When simplifying, you might combine like terms, perform basic arithmetic, or reduce fractions to their simplest form.

In our example, \(2(y+9)\), the simplification process began by using the distributive property to expand the expression, resulting in \(2y + 18\). No further simplification is needed in this case since there are no like terms to combine with \(2y\) or \(18\). Simplifying can make an expression easier to understand or make it more amenable to further manipulation such as factoring or solving for variable values.