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Chapter 3: Problem 6

Simplify each expression. $$5(y-8)-2 y$$

Short Answer

Expert verified
The simplified form of the expression \( 5(y-8) - 2y \) is \( 3y - 40 \).

Step by step solution

01

Apply the Distributive Property

The first operation to deal with is the multiplication of 5 across \( (y - 8) \). Applying distributive property, this will become \( 5y - 40 \). Accordingly, the expression will now be \( 5y - 40 - 2y \).
02

Combine Like Terms

Next, there will be simplification of the terms that contain the same variables. In this case, it is possible to combine \( 5y \) and \( -2y \) together. This simplification provides the new expression \( 5y - 2y - 40 \) which then simplifies to \( 3y - 40 \).

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Key Concepts

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

Understanding the Distributive Property
The distributive property is a fundamental concept in algebra, which allows us to multiply a single term across terms within parentheses. It makes expressions easier to manage and simplify. Say you have an expression like \( 5(y - 8) \). The distributive property tells us to multiply the 5 by each term inside the parentheses separately.
So, in detail, you multiply 5 by \( y \), and 5 by \( -8 \), which gives you \( 5y - 40 \).
This initial step of multiplying each term inside the parentheses is crucial to simplify complex expressions efficiently.
  • Think of the property as spreading the multiplication across all the terms inside the parentheses.
  • This technique is especially helpful in breaking down more complicated algebraic expressions.
Identifying and Combining Like Terms
Once you have applied the distributive property, the next step is to simplify the expression by combining like terms. Like terms are terms that contain the same variables raised to the same power. These can be combined to simplify algebraic expressions.
In our example, after distributing, the expression becomes \( 5y - 40 - 2y \). Both \( 5y \) and \( -2y \) are like terms because they contain the variable \( y \).
To simplify, you combine these like terms by adding or subtracting them. Here, you subtract \( 2y \) from \( 5y \), which results in \( 3y \).
  • Check the coefficient and the variable to ensure terms are alike.
  • Always perform the arithmetic operation between the coefficients of the like terms.
Completing the Simplification Process
After combining like terms, the expression can still be simplified if possible. The simplification process involves making the expression as concise as possible without changing its value.
In our exercise, we started with the expression \( 5(y - 8) - 2y \), applied the distributive property to get \( 5y - 40 - 2y \), and combined like terms to reach \( 3y - 40 \).
The final expression \( 3y - 40 \) is in its simplest form.
  • Always check if further combining is possible, though not in this scenario.
  • Remember that simplification not only makes expressions more manageable but also reveals equivalent expressions.

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