Problem 103 - Intermediate Algebra

Question

Simplify each expression. $$4.3(y+9)-8.1 y$$

Step-by-Step Solution

Answer

The simplified expression is \(-3.8y + 38.7\).

Step 1: Distribute the Coefficient

First, distribute the coefficient 4.3 to both terms inside the parentheses. This means you multiply 4.3 by each term separately: \[ 4.3 imes y + 4.3 imes 9 = 4.3y + 38.7 \]

Step 2: Rewrite the Expression

After distributing, rewrite the expression with the new terms: \[ 4.3y + 38.7 - 8.1y \]

Step 3: Combine Like Terms

Now, combine the like terms, which are the terms containing \( y \): \[ (4.3y - 8.1y) + 38.7 = -3.8y + 38.7 \]

Step 4: Write the Final Simplified Expression

The final expression after combining like terms is:\[ -3.8y + 38.7 \]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra. It allows you to simplify expressions in which a single term is being multiplied by terms that are inside parentheses. Here's how it works: when you have an expression like

• \( a(b + c) \)

you distribute \( a \) by multiplying it with both \( b \) and \( c \), the expression becomes

• \( ab + ac \)

Using this property, we can break down expressions into simpler parts that are easier to manage.
In our original exercise, the expression was

• \( 4.3(y+9) \)

By using the distributive property, we multiplied \( 4.3 \) with \( y \) and \( 9 \), resulting in

• \( 4.3y + 38.7 \)

This allowed us to remove the parentheses and proceed to simplify the expression further.

Like Terms

When working with algebraic expressions, recognizing and combining like terms is an essential skill. Like terms are terms in an expression that contain the same variable raised to the same power. For example,

• \( 4.3y \) and \( -8.1y \)

are like terms because they both contain the variable \( y \).
In order to combine them, you simply add or subtract their coefficients, which are the numerical parts of the terms. The variable part stays the same. In our example, we combined

• \( 4.3y \) and \( -8.1y \)

by calculating

• \( 4.3 - 8.1 = -3.8 \)

Thus, the combined like terms were

• \( -3.8y \)

Understanding like terms helps simplify expressions and solve equations more efficiently.

Simplifying Expressions

Simplifying expressions means reducing them to their most basic form while still representing the same value or relation. This involves applying properties like the distributive property, combining like terms, and performing arithmetic operations
In our exercise, after distributing and combining like terms, we reached the simplified expression

• \( -3.8y + 38.7 \)

This is considered simplified because there are no more parentheses, all like terms have been combined, and it contains the fewest terms possible.
To simplify any algebraic expression, follow these general steps:

• Use the distributive property to eliminate any parentheses
• Identify and combine all like terms
• Perform any basic arithmetic if necessary

Simplifying makes expressions easier to work with and is a critical step in solving more complex equations.