Problem 33 - Prealgebra

Question

Simplify expression. \(6 x+4-7 x\)

Step-by-Step Solution

Answer

The simplified expression is \(-x + 4\).

Step 1: Identify Like Terms

In the expression \(6x + 4 - 7x\), we have terms that are coefficients of \(x\). These terms are \(6x\) and \(-7x\). Identify these as like terms because they both contain the variable \(x\).

Step 2: Combine Like Terms

Combine the coefficients of the like terms \(6x\) and \(-7x\). This is done by subtracting the coefficients: \(6 - 7 = -1\). Therefore, the expression becomes \(-1x + 4\).

Step 3: Simplify the Coefficients

Any term with a coefficient of \(-1\) can simply be written as \(-x\). Therefore, \(-1x + 4\) simplifies to \(-x + 4\).

Key Concepts

Understanding Like Terms

In algebra, recognizing and working with like terms is essential for simplifying expressions. Like terms are terms in an expression that have the same variable raised to the same power. For example, in the expression \(6x + 4 - 7x\), the terms \(6x\) and \(-7x\) are considered like terms because they both contain the variable \(x\) raised to the first power. On the other hand, the number \(4\) is not a like term with \(6x\) or \(-7x\) because it lacks the variable \(x\).
Combining like terms is key to simplifying expressions. To do so, focus on the coefficients—the numbers in front of the variables. By adding or subtracting these coefficients, you can consolidate the expression, making it easier to understand and solve.
Remember: Only terms with exactly the same variable component can be combined. This is why before combining terms, you should carefully identify which ones are truly like terms.

Decoding Coefficients

Coefficients are the numerical parts of terms that contain variables. In the expression \(6x + 4 - 7x\), \(6\) and \(-7\) are the coefficients of the terms that contain the variable \(x\). The coefficient shows how many times the variable is multiplied.
When simplifying, the process typically involves combining coefficients of like terms. This means adding or subtracting the coefficients while keeping the variable part unchanged. For instance, when we have \(6x\) and \(-7x\), we subtract the coefficients, because one coefficient is positive and the other negative: \(6 - 7 = -1\). This results in the combined term \(-1x\).
If the coefficient is \(1\) or \(-1\), it is a simplifying rule to write just the variable, such as \(x\) or \(-x\). This practice reduces extra clutter and makes the expression neater and quicker to interpret.

Exploring Variable Expressions

A variable expression consists of numbers, variables, and arithmetic operations. Variables, often represented by letters like \(x\), act as placeholders that can assume various values depending on the context.
The expression \(6x + 4 - 7x\) is a variable expression, as it contains numbers (\(6\) and \(-7\)), a constant (\(4\)), and a variable part (\(x\)).
Understanding how to simplify variable expressions involves reducing them to their simplest form without changing their value. This aids in solving equations and understanding relationships between variables and numbers.
To simplify a variable expression:

• Identify and combine like terms.
• Perform the arithmetic operations, paying careful attention to the signs of coefficients.
• Present the simplified result in a clear format, such as \(-x + 4\), which is the simplest form of the original expression.

Recognizing the components of a variable expression is also crucial because it enables you to solve algebraic equations more effectively by isolating the variable correctly.