Problem 19 - Algebra and Trigonometry

Question

Find the sum, difference, or product. \((12 x-7)-(5 x-12)\)

Step-by-Step Solution

Answer

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

Step 1: Distribute the Negative Sign

First, distribute the negative sign across the terms inside the parenthesis in the second expression. This changes \((5x - 12)\) to \(-5x + 12\).

Step 2: Combine Like Terms

Rewrite the expression as \((12x - 7) + (-5x + 12)\). Now, combine the like terms. There are two groups of like terms: the terms involving \(x\) and the constant terms. Combine the \(x\) terms: \(12x - 5x = 7x\). Combine the constant terms: \(-7 + 12 = 5\).

Step 3: Write the Simplified Expression

Combine the results from Step 2 to write the simplified expression: \(7x + 5\). This is the result of \((12x - 7) - (5x - 12)\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental concept in algebra that makes expressions simpler and easier to work with. When you encounter terms in an algebraic expression, it's important to identify which ones can be combined based on their variables and coefficients.

The key principle is to only combine terms that have exactly the same variable raised to the same power. For example, in the expression \(12x - 5x + 7\), \(12x\) and \(-5x\) are like terms because they both involve the same variable, \(x\). The numbers in front, 12 and -5, are called coefficients.

• Combine the coefficients of like terms to simplify: \(12x - 5x\).
• This results in \(7x\), as you subtract 5 from 12.
• Terms like \(+7\) do not have the variable \(x\) and are considered constants, thus they cannot be combined with \(12x\).

Combining like terms reduces the complexity of expressions and prepares them for further manipulation.

Distributive Property

The distributive property is a useful tool in algebra that helps to simplify expressions, especially when dealing with subtraction or multiplication of grouped terms. This property states that \(a(b + c) = ab + ac\). It allows us to distribute or "spread out" a multiplication across terms within parentheses.

In the expression \((12x - 7) - (5x - 12)\), the negative sign before the parenthesis \((5x - 12)\) needs to be distributed to each term inside.

• This means changing \(5x\) to \(-5x\) and \(12\) to \(-12\).
• The expression becomes \(12x - 7 + (-5x + 12)\), making it ready for the next step in simplifying.

The distributive property is crucial for handling expressions with multiple terms and allows us to proceed with operations like combining like terms.

Simplifying Expressions

Simplifying expressions involves reducing an algebraic expression to its simplest form. This makes it easier to understand and work with, particularly in solving equations. After utilizing the distributive property and combining like terms, we reach a streamlined version of the expression.

In the exercise example, after simplifying \((12x - 7) - (5x - 12)\), you combine like terms to get \(7x + 5\).

• This new expression, \(7x + 5\), is much simpler and contains no parentheses.
• By simplifying, you eliminate unnecessary complex structures, making it easier for further algebraic manipulation or substitution.

Remember, a simplified expression is key in problem-solving as it often represents the most straightforward path to the solution. It is also essential for ensuring clarity and avoiding errors in calculations.