Problem 70 - Prealgebra

Question

Simplify each of the following expressions as much as possible. $$5(y+3)+7$$

Step-by-Step Solution

Answer

The simplified expression is \(5y + 22\).

Step 1: Distribute the Constant

Distribute the constant 5 across the terms in the parentheses. This involves multiplying 5 by both terms inside the parentheses: \(5 \times y + 5 \times 3\).

Step 2: Multiply Each Term Inside Parentheses

Calculate the multiplication for each term: \(5 \times y = 5y\) and \(5 \times 3 = 15\). Thus, the expression is transformed to \(5y + 15 + 7\).

Step 3: Combine Like Terms

Combine the constant terms in the expression. Add the numbers 15 and 7 together to consolidate them: \(5y + 15 + 7 = 5y + 22\).

Step 4: Final Simplified Expression

The expression is fully simplified to \(5y + 22\), as no further like terms are left to combine.

Key Concepts

Simplification of Expressions

Simplification of algebraic expressions involves finding an equivalent expression that is most condensed. This process helps in understanding and solving problems efficiently. To begin simplifying, first look for opportunities to perform operations like multiplication and addition on any numbers or variables. This clears up the expression and makes it easier to read. The primary goals during simplification involve applying arithmetic operations where possible and removing any unnecessary symbols or terms. In our example, the expression given was: 5(y+3)+7 The aim is to perform operations that reduce this expression to a simpler form, eventually leading to a clean and concise result like 5y + 22.

Distributive Property

The distributive property is a fundamental rule in algebra. It's a way to multiply a single term by more than one term inside a set of parentheses. This property states that when you have an expression of the form a(b + c), you can distribute the multiplication of 'a' across the terms inside the parentheses. This means:\[a(b + c) = ab + ac\]In our exercise, the distributive property was used to multiply the number 5 with both 'y' and 3 within the parentheses (y+3):

• 5 multiplied by y results in 5y
• 5 multiplied by 3 results in 15

This simplifies that part of the expression to 5y + 15. The distributive property allows us to handle expressions efficiently by breaking them down into manageable parts.

Combining Like Terms

Combining like terms is the next step after you've used properties like distribution to simplify the expression. Like terms are terms that have the same variable raised to the same power. For example, in the expression 3x + 4x, both terms have the variable 'x' and can be combined.In our exercise, after distributing and simplifying, the expression was:\[5y + 15 + 7\]Here, the like terms are the constant numbers 15 and 7. Combining them involves adding these numbers together:

• 15 and 7 are combined to make 22

By combining like terms, the expression becomes easier to manage and, in this case, results in the final simplified form:\[5y + 22\]This ensures the expression is as simple as possible without changing its value.