Problem 18 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ 3(x-1)+4(x+2) $$

Step-by-Step Solution

Answer

The simplified expression is \(7x + 5\).

Step 1: Distribute Multiplication

First, distribute the multiplication over the terms inside each of the parentheses. For the term \(3(x-1)\), multiply 3 by both \(x\) and \(-1\) to get \(3x - 3\). For the term \(4(x+2)\), multiply 4 by both \(x\) and \(2\) to get \(4x + 8\). The expression now becomes \(3x - 3 + 4x + 8\).

Step 2: Combine Like Terms

Now, combine the like terms in the expression \(3x - 3 + 4x + 8\). First, combine the \(x\)-terms: \(3x + 4x = 7x\). Next, combine the constant terms: \(-3 + 8 = 5\). This simplifies the expression to \(7x + 5\).

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that helps us manage expressions involving both parentheses and multiplication. This property states that multiplying a single term by terms within parentheses can be achieved by multiplying the single term with each term inside the parentheses. It's like hand-delivering cookies from a large jar to different boxes. For example, when you have an expression like \(3(x - 1)\), you'll multiply the 3 by both \(x\) and \(-1\), resulting in \(3x\) and \(-3\). Similarly, in \(4(x + 2)\), the 4 is multiplied by both \(x\) and 2, leading to \(4x + 8\). By distributing each outside number across the terms inside the parentheses, we simplify our calculations and keep track of every term neatly.

Combining Like Terms

Combining like terms is akin to sorting socks; you group similar items together to make sense of them. Like terms in algebra are terms that have the exact same variables raised to the same power. These are the terms you can legally add or subtract. In the expression \(3x + 4x - 3 + 8\), we group our like terms together:

• Combine the \(x\) terms: \(3x + 4x\) which simplifies to \(7x\)
• Combine the constant terms: \(-3 + 8\) which results in 5

This gives us a simplified expression of \(7x + 5\). Grouping similar terms helps us tidy up our expressions and make them easier to work with.

Algebraic Operations

Algebraic operations encompass actions like addition, subtraction, multiplication, and division performed on algebraic expressions. These operations are the essential tools for manipulating and simplifying expressions, enabling us to solve equations or simply make them more readable.In the problem \(3(x-1) + 4(x+2)\), we applied algebraic operations in a few steps:

• First, use the distributive property to eliminate parentheses by multiplying across the terms.
• Next, simplify the expression by combining like terms, making sure to only add or subtract terms that have identical variable parts.

Understanding and correctly applying algebraic operations ensures we can break down complex expressions into manageable pieces, leading us to a much simpler form like \(7x + 5\). This stepwise approach is key in maximizing the clarity and simplicity of mathematical expressions.