Problem 53 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ y+5(y+6) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: \(y + 5(y + 6)\). Answer: \(6y + 30\)

Step 1: Apply the distributive law

Distribute the "\(5\)" to both the "\(y\)" and the "\(6\)" inside the parentheses. To do this, multiply \(5\) by each term inside the parentheses: $$ y + 5(y + 6) = y + (5 * y) + (5 * 6) $$

Step 2: Multiply the numbers with the variables

Now, perform the multiplication involved in the expanded expression: $$ y + (5 * y) + (5 * 6) = y + 5y + 30 $$

Step 3: Combine like terms

Finally, combine the like terms (the terms containing "\(y\)") to simplify the expression: $$ y + 5y + 30 = 6y + 30 $$ So, the simplified algebraic expression is: $$ 6y + 30 $$

Key Concepts

Distributive Law

Understanding the distributive law—also known as the distributive property—is critical for manipulating and simplifying algebraic expressions. This property illustrates how to distribute a single term over terms inside parentheses. Consistently applying the distributive law can help avoid mistakes when dealing with complex expressions in algebra.

Consider the expression: \( y + 5(y + 6) \). To apply the distributive law, you multiply the term outside the parentheses (in this case, 5) by each of the terms inside the parentheses. Mathematically, this is represented as: \( 5 \times y \), and \( 5 \times 6 \). As a result, the expression becomes \( y + 5y + 30 \), where the distributive law has effectively been used to eliminate the parentheses by spreading the multiplication.

Combining Like Terms

Once the distributive law has been applied, the next step is combining like terms. This process involves merging terms that have the exact same variable part. It's important to understand that only the coefficients of like terms can be combined; this is because they represent the same quantity of something, just in a greater or lesser amount.

For instance, in the expression \( y + 5y + 30 \), \( y \) and \( 5y \) are like terms since they both contain the variable \( y \). They can easily be combined by adding their coefficients (1 and 5) together, giving us \( 6y \). Therefore, combining like terms of the original expression simplifies it to \( 6y + 30 \) , a much cleaner and easier expression to work with.

Algebraic Simplification

The final goal in solving algebraic expressions is to simplify them to their most basic form, a process known as algebraic simplification. After applying the distributive law and combining like terms, the expression should be as compact and straightforward as possible. It makes the expressions clearer and often easier to use in equations or further calculations.

Take our example \( 6y + 30 \). It's already quite simplified, but suppose there was a common factor in the terms; you would factor that out or divide to further simplify the expression. Algebraic simplification is the act of minimizing an algebraic expression without changing its value—an essential skill in mathematics that provides a strong foundation for solving equations and understanding complex topics.