Problem 25 - Elementary Algebra

Question

Simplify. $$ -8(8 x-3)-7 $$

Step-by-Step Solution

Answer

The simplified expression is -64x + 17.

Step 1: Distribute the -8

Multiply -8 by each term inside the parentheses: -8 times (8x) and -8 times (-3). Do this step by step: -8 times 8x gives you -64x. -8 times -3 gives you +24. Therefore, the expression now becomes: -64x + 24 - 7.

Step 2: Combine Like Terms

Now, combine the constant terms: +24 and -7. 24 - 7 equals 17. Thus, the simplified expression is: -64x + 17.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to remove parentheses from expressions. When applying the distributive property, you take a single term outside the parentheses and multiply it by each term inside the parentheses.
This means multiplying the term outside the parentheses with each item inside. In the given exercise, it involves taking the
-8 and multiplying it by everything inside the parentheses:

• -8 is multiplied by 8x, giving us \(-64x\).
• -8 is also multiplied by \(-3\), resulting in \(+24\).

By doing so, you effectively distribute \(-8\) across all terms inside the parentheses, simplifying the expression further. This step helps to break down complex expressions into simpler ones, making the process of solving equations or further simplifying them a lot easier. Understanding the distributive property is crucial because it helps eliminate parentheses, which can sometimes make expressions appear more complex than they really are.

Combining Like Terms

Once you’ve used the distributive property, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. Only the numerical coefficients of these terms may differ. In this exercise, after applying the distributive property, the expression
-64x + 24 - 7 is obtained.

• Identify and group like terms. Here, -64x is the term with the variable, and the constants are 24 and -7.
• Combine the numbers, or constant terms, together: \(24 - 7\) gives \(17\).

The combining like terms step is vital because it further simplifies the expression, reducing it to its most simple form. It makes expressions easier to read and understand and sets the stage for solving equations if any are involved.

Algebraic Simplification

Algebraic simplification is the process of transforming an expression into its simplest form. This involves performing operations such as using the distributive property and combining like terms as seen in the previous sections.
The goal is to make the expression as simple and concise as possible, reducing clutter and complexity. For the problem \(-8(8x-3)-7\), the steps involved:

• First, use the distributive property to eliminate parentheses.
• Then, proceed to combine like terms, specifically focusing on constants.
• The resulting simplified expression is \(-64x + 17\).

By practicing algebraic simplification, you build the foundation for solving more complex equations and delve deeper into algebra. Simplification is an essential algebraic skill that transforms complicated expressions into manageable ones, making problem-solving faster and less error-prone.