Problem 105 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (y+3 a)(2 y+a) $$

Step-by-Step Solution

Answer

Question: Using the distributive property, simplify the algebraic expression: (y + 3a)(2y + a). Answer: 2y^2 + 7ay + 3a^2

Step 1: Apply the distributive property

To use the distributive property, we start with the first term of the first parentheses, which is y, and multiply it by each term in the second parentheses. Then do the same thing with the second term in the first parentheses, which is 3a.

Step 2: Multiply the terms

So the multiplication process will be like this: $$ (y)(2y) + (y)(a) + (3a)(2y) + (3a)(a) $$

Step 3: Simplify the terms

Next, we need to simplify each term by carrying out the multiplications: $$ 2y^2 + ay + 6ay + 3a^2 $$

Step 4: Combine like terms

Now, we need to combine the like terms, which are the middle terms ay and 6ay. Adding these terms together, we get 7ay. Thus, the simplified expression is: $$ 2y^2 + 7ay + 3a^2 $$ The final simplified algebraic expression for the problem is: $$ (2y^2 + 7ay + 3a^2) $$

Key Concepts

Algebraic Expressions

Algebraic expressions are a crucial part of mathematics and appear frequently in different areas, from simple equations to complex functions. They are made up of numbers, variables, and operators, such as addition, subtraction, multiplication, and division. An important aspect of algebraic expressions is that they can represent real-world problems where exact values may not be known, allowing for flexible and dynamic problem-solving approaches.
Algebraic expressions are foundational because they form the basis for solving equations and understanding relationships between variables. The expression \((y+3a)(2y+a)\) represents a combination of terms where multiplication is conducted across two binomials. These aspects make understanding algebraic expressions essential as they help to recognize patterns, make predictions, and express quantities in a concise way.

Simplification

Simplification is the process of making an algebraic expression easier to understand or work with by reducing it to its simplest form. It involves systematically performing operations and combining terms where possible. This process often begins with applying the distributive property, which involves multiplying each term inside a pair of parentheses by a term outside the parentheses, or simply expanding binomials such as \((y+3a)(2y+a)\).

Simplifying algebraic expressions ensures they are as concise as possible, reducing potential errors in solving equations and making it easier to work with larger expressions. At its core, simplification yields an expression that is logically equivalent to the original but is more straightforward to interpret and eventually solve.

Combining Like Terms

Combining like terms is an essential step in the simplification process of algebraic expressions. "Like terms" are terms within an expression that have the same variable raised to the same power. In the case of simplifying \(2y^2 + ay + 6ay + 3a^2,\) terms like \(ay\) and \(6ay\) are identified as like terms due to them having the same variable and power.

By combining these terms, you efficiently consolidate similar parts of an expression into a single term, leading to a cleaner, more streamlined result. For the expression above, combining \(ay\) and \(6ay\) results in \(7ay,\) simplifying the expression into \(2y^2 + 7ay + 3a^2.\)

This step is crucial for producing expressions that are easier to evaluate, solve, or further manipulate.