Problem 24 - Algebra and Trigonometry

Question

Find the sum, difference, or product. \(3(x-1)+4(x+2)\)

Step-by-Step Solution

Answer

The expression simplifies to \(7x + 5\).

Step 1: Expand the Expressions

Distribute the numbers 3 and 4 into their respective parentheses:- For the first term: \(3(x-1) = 3 \cdot x - 3 \cdot 1 = 3x - 3\).- For the second term: \(4(x+2) = 4 \cdot x + 4 \cdot 2 = 4x + 8\).

Step 2: Combine Like Terms

Add the like terms together:- Combine \(3x\) and \(4x\) to get \(7x\).- Combine \(-3\) and \(8\) to get \(5\).Thus, the expression becomes \(7x + 5\).

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental principle in algebra that allows us to simplify expressions. In essence, it states that when you multiply a number by a sum or difference, you distribute the multiplication across each term inside the parentheses. This principle can be expressed as:

• For any numbers a, b, and c: \(a(b + c) = ab + ac\)
• Similarly, \(a(b - c) = ab - ac\)

Applying the distributive property helps break down more complex expressions into simpler parts, making calculations easier to manage. In our original exercise, the distributive property helped us to transform \(3(x-1) + 4(x+2)\) by multiplying each term inside the parentheses by the numbers outside. As a result, we got \(3x - 3\) and \(4x + 8\). These steps pave the way for the next process of simplifying the expression further.

Combining Like Terms for Clarity

Combining like terms is a vital skill when working with polynomial expressions. This process involves adding or subtracting terms that have the same variable and exponent. It is essential because it simplifies expressions and makes them easier to understand and work with. When you combine like terms, you ensure each variable is grouped together, which reduces complexity:

• For example, in the expression \(3x + 4x\), since both terms have the variable \(x\) raised to the same power, they are like terms and can be simplified to \(7x\).
• Similarly, constant terms like \(-3\) and \(8\) can be combined to yield \(5\).

In our solved problem, combining \(3x\) with \(4x\) gives us \(7x\), and combining \(-3\) with \(8\) results in \(5\). This means we've condensed the expression to \(7x + 5\), providing a simpler, cleaner version of the original expression.

Simplifying Algebraic Expressions

Algebraic simplification involves making an expression as concise as possible without changing its value. This process often includes using the distributive property and combining like terms. By simplifying expressions, we can more easily interpret and solve mathematical problems.A well-simplified expression is typically:

• Free of unnecessary parentheses
• Contains only the minimal number of terms
• Organized, generally from the highest power of the variable down to the constants

In our exercise, we started with the expression \(3(x-1) + 4(x+2)\). By distributing and then combining like terms, we simplified it to \(7x + 5\). This final expression represents a tidier and more manageable form—it shows the sum of products explicitly and eliminates any extraneous components. Simplification is not just about making things neat; it's crucial for solving equations efficiently and accurately.