Problem 86 - Introductory Algebra for College Students

Question

Simplify: \(3[7 x-2(5 x-1)] .\) (Section \(1.8,\) Example 11 )

Step-by-Step Solution

Answer

The simplified expression is \(-9x + 6\).

Step 1: Distribute the Inner Parentheses

Distribute the multiplication across each term inside the inner parentheses. So, \(2(5x - 1)\) becomes \(10x - 2\). This gives us \(3[7x - (10x - 2)]\).

Step 2: Remove the Brackets

To remove the brackets, take care of the sign in front of the brackets. Multiplies both terms inside the brackets by -1 to get \(3[7x - 10x + 2]\)

Step 3: Simplify Inside the Brackets

Combine like terms inside the brackets giving \(3[-3x + 2]\). Again distribute 3 to each term inside the brackets. This yields \(-9x + 6\).

Step 4: Provide the Simplified Expression

The simplified form of the original expression is \(-9x + 6\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves various steps to make an algebraic expression more manageable and easier to understand. A simplified expression is usually free from parentheses and has all like terms combined for clarity. In our example, to simplify,

• we initiate by resolving inner expressions within any parentheses.
• Apply any necessary operations, such as addition, subtraction, multiplication, or division, as guided by the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

By resolving what's inside the parentheses first, we ensure the expression adheres to mathematical rules. Our goal is a cleaner, less complex expression. Good simplification provides a clear pathway to the solution.
In academic settings, simplifying expressions is a critical skill that facilitates understanding of more complex algebraic structures. Practicing simplification helps you to see relationships and patterns among terms, making solving algebraic equations easier down the line.

Distributive Property

The distributive property is a fundamental property of arithmetic and algebra that allows us to simplify expressions. It states that for all numbers, \[ a(b+c) = ab + ac \]This means we distribute the multiplication of one term across the terms added (or subtracted) inside the parentheses. For example, if you have the expression \[ 2(5x - 1) \]this can be expanded by distributing the 2 to each term inside the parentheses, resulting in \[ 10x - 2 \].In our specific problem, the distributive property was applied twice, first to remove the inner parentheses and then to further expand the remaining expression. Mastering the distributive property is essential because it enables you to break down complex expressions into simpler, more solvable parts.
This approach is helpful in simplifying polynomials or solving equations, where expressions generally appear nested in layers of parentheses.

Combining Like Terms

Combining like terms is a process used in algebra to simplify expressions. Like terms are terms that contain the same variable raised to the same power. Their coefficients can vary, but the variable part must be identical. For example, \[ 7x \text{ and } -10x \]can be combined because they both contain the variable \[ x \].When simplifying an expression, identify and combine all like terms to shrink the equation to its simplest form. In the solution provided, combining \[ 7x - 10x \]inside the brackets resulted in \[ -3x \].By grouping and simplifying these terms, you reduce the complexity of the expression, making it much easier to handle and solve.How do we combine them? Only adjust the coefficients while maintaining the variable and power. Understanding and applying this concept relieves much of the intimidation factor often associated with algebraic expressions and is a necessary step towards solving equations effectively.