Problem 77 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(10[0.3(5-3 x)]\)

Step-by-Step Solution

Answer

15 - 9x

Step 1: Distribute inside the parentheses

First, distribute the 0.3 inside the parentheses of the expression. This means multiplying 0.3 by each term inside the parentheses.\[0.3(5 - 3x) = 0.3 \times 5 - 0.3 \times 3x\]This becomes:\[1.5 - 0.9x\]

Step 2: Multiply outside the brackets

Next, distribute the 10 outside the brackets through the expression obtained in step 1. This means multiplying 10 by each term inside the brackets.\[10(1.5 - 0.9x) = 10 \times 1.5 - 10 \times 0.9x\]This simplifies to:\[15 - 9x\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra and is used to simplify expressions. It states that multiplying a sum by a number gives the same result as multiplying each addend by that number and then adding the products.
In mathematical terms, this is written as:
\[a(b + c) = ab + ac\].
In the exercise, to simplify the expression \(10[0.3(5-3 x)]\), the distributive property is used twice:

• First, the 0.3 is distributed inside the parentheses: \[0.3(5 - 3x) = 0.3 \times 5 - 0.3 \times 3x = 1.5 - 0.9x\].
  
• Then, the 10 is distributed through the result from the first distribution: \[10(1.5 - 0.9x) = 10 \times 1.5 - 10 \times 0.9x = 15 - 9x\].
  

This property makes it easier to solve and simplify algebraic expressions.

Multiplication in Algebra

Multiplication is one of the basic operations in algebra and is essential for solving many types of equations and expressions.
When multiplying terms, it's important to follow the rules of arithmetic and the properties of operations, such as the distributive property.
In the given exercise, multiplication is used in two steps to simplify the expression:

• First, we multiply 0.3 by each term inside the parentheses \((5 - 3x)\) to get \(1.5 - 0.9x\).
  
• Second, we multiply each term of the resulting expression by 10 to finalize the simplification: \(10 \times 1.5\) and \(10 \times -0.9x\), resulting in \(15 - 9x\).
  

Understanding multiplication and its interaction with other algebraic operations like addition, subtraction, and distribution is crucial for mastering algebra.

Linear Expressions

Linear expressions are algebraic expressions where the variable(s) are of the first degree, meaning they are not raised to any exponent other than one.
They often appear in the form \(ax + b\), where \(a\) and \(b\) are constants.
In the exercise, the final simplified expression \(15 - 9x\) represents a linear expression, because the variable \(x\) is of the first degree.
Such expressions can be easily plotted on a graph as a straight line.
Key points about linear expressions include:

• They have one variable term \(ax\) and a constant term \(b\).
  
• The coefficient \(a\) indicates the slope of the line when graphed.
  
• The constant \(b\) indicates where the line intersects the y-axis.
  
• Simplifying linear expressions involves combining like terms and using properties like distribution.
  

Understanding linear expressions is important as they are foundational for learning more complex algebra and calculus.