Problem 58 - Elementary Algebra

Question

\((2 x-9)-(3 x-4)-(7 x+8)\)

Step-by-Step Solution

Answer

-8x - 13

Step 1: Expand the expression

Remove the parentheses by distributing the negative sign where necessary. The expression becomes (2x - 9) - (3x - 4) - (7x + 8)

Step 2: Apply the distributive property

Distribute the negative sign inside the parentheses: (2x - 9) - 3x + 4 - 7x - 8

Step 3: Combine like terms

Combine the terms with 'x' together and the constant terms together: 2x - 3x - 7x - 9 + 4 - 8

Step 4: Simplify

Combine the terms: (2x - 3x - 7x) + (-9 + 4 - 8) This simplifies to: -8x - 13

Key Concepts

Distributive Property

The distributive property is a critical algebraic rule used to simplify expressions. This property states that multiplying a number by a sum or difference is the same as multiplying each addend separately and then adding or subtracting the products. In algebra, it is written as:
\(a(b + c) = ab + ac\).
Let's see this concept in action using the given exercise. When we have \((2x - 9) - (3x - 4) - (7x + 8)\), distributive property tells us to distribute the negative sign to each term inside the parentheses:
distributing \(-(3x - 4)\) results in \(-3x + 4\), and distributing \(-(7x + 8)\) results in \(-7x - 8\).
This simplifies our expression to: \(2x - 9 - 3x + 4 - 7x - 8\).
See how removing parentheses makes it easier to combine like terms!

Simplifying Expressions

Simplifying expressions means making them as simple as possible by combining like terms and using algebraic rules. After applying the distributive property, our next job is to simplify. In the expression \(2x - 9 - 3x + 4 - 7x - 8\), this involves:

• Combining the 'x' terms: \(2x - 3x - 7x\).
• Combining the constant terms: \(-9 + 4 - 8\).

Combining the 'x' terms gives us: \(2x - 3x - 7x = -8x\). And combining the constant terms gives us: \(-9 + 4 - 8 = -13\).
Simplifying involves these steps to streamline the expression into a more manageable form: \(-8x - 13\).

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (like plus and minus). It's different from a simple arithmetic expression because it can include variables represented by symbols like 'x' or 'y'.
Consider the expression we started with: \((2x - 9) - (3x - 4) - (7x + 8)\). This expression:

• Includes terms with 'x' (the variable) and constants (plain numbers).
• Utilizes operators like subtraction to separate the terms.

In algebra, combining like terms and applying properties such as the distributive property makes expressions easier to work with. By following the steps manually, students learn how to translate complex expressions into simpler forms. The final simplified form \(-8x - 13\) is much easier to interpret and use in further calculations.