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Chapter 1: Problem 95

Simplify. $$ 2 x-7 x-(4 x-6) $$

Short Answer

Expert verified
-9x + 6

Step by step solution

01

- Distribute negative sign

Rewrite the expression by distributing the negative sign into the parentheses: \(2x - 7x - 4x + 6\).
02

- Combine like terms

Combine all the terms with \(x\): \(2x - 7x - 4x\). This simplifies to \(-9x\).
03

- Simplify constants

Add the constant term: \(6\).
04

- Combine final expression

The simplified expression is \(-9x + 6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is essential when dealing with algebraic expressions. It states that multiplying a sum by a number is the same as multiplying each addend by the number, then adding the products. In our exercise, we use the distributive property to handle the negative sign in front of the parentheses.

For the expression: \(2x - 7x -(4x - 6)\), we need to distribute the negative sign through the terms inside parentheses. This changes \( -(4x - 6) \) to \( -4x + 6 \). So, the expression becomes:
\(2x - 7x - 4x + 6\).

Distributing first helps keep track of negative signs and ensures accurate calculation in subsequent steps.
Combining Like Terms
When simplifying algebraic expressions, combining like terms is crucial. Like terms are terms that contain the same variable raised to the same power. This step helps in consolidating the expression into a simpler form.

In our example, after applying the distributive property, we have:
\(2x - 7x - 4x + 6\). Here, the terms with \(x\) are like terms. We need to combine them:
\(2x - 7x - 4x = -9x\).

Now the expression looks much simpler:
\(-9x + 6\).

Combining like terms simplifies further calculations and makes the expression manageable.
Algebraic Simplification
Algebraic simplification removes unnecessary complexity from an expression and finds its simplest form. The process involves using properties of algebra like the distributive property and combining like terms. This systematic simplification makes the expression easier to understand and work with.

In our example, after combining like terms, we have \(-9x + 6\). There are no more like terms to combine or parentheses to distribute.

Thus, the expression \(2x - 7x - (4x - 6)\) simplifies to:
\(-9x + 6\).

Algebraic simplification is about transformation and reducing an expression to its most straightforward form, making it more efficient for further evaluation or solving equations.

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