Problem 62 - Algebra A Combined Function

Question

Perform each indicated operation. Don't forget to simplify if possible. Subtract \(4 x-7\) from \(12+x\)

Step-by-Step Solution

Answer

The simplified expression is \( 19 - 3x \).

Step 1: Identify the Expressions

We have two expressions: the expression to subtract from is \( 12 + x \) and the expression to subtract is \( 4x - 7 \).

Step 2: Write the Subtraction Expression

To subtract \( 4x - 7 \) from \( 12 + x \), we set up the expression: \((12 + x) - (4x - 7)\).

Step 3: Distribute the Negative Sign

Distribute the negative sign across the second expression: \( (12 + x) - 1(4x - 7) = 12 + x - 4x + 7\).

Step 4: Combine Like Terms

Combine the like terms: \( (12 + 7) + (x - 4x) = 19 - 3x \).

Step 5: Write the Simplified Expression

The simplified result of the subtraction is \( 19 - 3x \).

Key Concepts

Simplification

Simplification in algebra refers to the process of turning a complex expression into a simpler or more easily manageable form. This process is crucial in solving equations or performing arithmetic operations, such as subtraction, as it allows you to clearly see the relationships between variables and constants. When dealing with expressions like \((12 + x) - (4x - 7)\), it's important to convert it into its simplest form to make further operations straightforward. Simplification often involves eliminating any unnecessary terms or combinations of terms through mathematical operations. It's akin to cleaning up a messy room: you want to rearrange items so everything is neat and tidy. Key elements to look for during simplification include:

• Removing parentheses by distributing any numerical factors.
• Combining similar terms, like terms which involve the same variable raised to the same power.
• Ensuring all terms are in their lowest terms or simplest form possible.

Applying proper simplification techniques ensures clarity, thus making any additional mathematical procedures less daunting.

Distributing Negative Signs

Distributing negative signs is a crucial step when dealing with subtraction in algebra. When you subtract an expression, you essentially distribute a negative sign across all terms within the parentheses. This means each term inside the parentheses should be multiplied by \(-1\). This reverses their signs, which could be either changing a positive to a negative or vice versa.For example, in the expression \((12 + x) - (4x - 7)\), distributing the negative sign results in:\[12 + x - 4x + 7\]Here's how it works:

• The \(-1\) is multiplied by \(4x\), switching \(4x\) to \(-4x\).
• The \(-1\) is also multiplied by \(7\), changing \(-7\) to \(+7\).

This simplifies the expression by turning a subtraction into an addition where needed, enabling easier solving processes. Understanding and correctly applying this concept prevents errors and aids in reaching the correct simplified expression.

Combining Like Terms

Combining like terms is a method used in algebra to simplify expressions by merging terms that have identical variable parts. This technique is part of simplifying complex expressions, making it easier to solve or further manipulate them.In the given example, once you have distributed the negative and have the expression \(12 + x - 4x + 7\), your goal is to identify and combine like terms.Here's what combining like terms looks like:

• Identify terms with the same variables: \(x\) and \(-4x\).
• Combine them as follows:\(x - 4x = -3x\).
• Do the same for constant numbers: \(12\) and \(7\), for which \(12 + 7 = 19\).

Once like terms are combined, the algebraic expression is simplified to \(19 - 3x\). This consolidated expression is easier to work with and allows you to clearly see the coefficient of each variable, setting up for further manipulations or solutions. Remember, combining like terms ensures that every variable is accounted for and expressed in its simplest form possible.