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Chapter 1: Problem 79

Simplify each expression. $$ -7.5(2 y+4)-2.9(3 y-6) $$

Short Answer

Expert verified
-23.7y - 12.6

Step by step solution

01

- Distribute the constants

Multiply each term inside the parentheses by the constants outside. For the first part: \[ -7.5(2y + 4) = (-7.5 \times 2y) + (-7.5 \times 4) = -15y - 30 \] For the second part: \[ -2.9(3y - 6) = (-2.9 \times 3y) + (-2.9 \times -6) = -8.7y + 17.4 \]
02

- Combine like terms

Add the coefficients of the like terms, which are the terms with 'y' and the constant terms: \[ -15y - 8.7y - 30 + 17.4 = (-15y - 8.7y) + (-30 + 17.4) = -23.7y - 12.6 \]

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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 a fundamental concept in algebra. It allows you to simplify expressions where you have a constant outside of the parentheses. This property states that you multiply the constant by each term inside the parentheses separately before performing any addition or subtraction.
For example, in the problem \ -7.5(2y + 4), you distribute -7.5 across both 2y and 4:
\[ -7.5 \times 2y = -15y, \text{ and } \ -7.5 \times 4 = -30 \]
The distributive property helps break down more complicated expressions into simpler pieces that are easier to work with.
Combining Like Terms
Once you have used the distributive property, you need to combine like terms. Like terms are terms that contain the same variable raised to the same power. In our example: \( -7.5(2y + 4) - 2.9(3y - 6) \), after applying the distributive property, we get:
\( -15y - 30 - 8.7y + 17.4 \).
Now, you identify the like terms. Here, \( -15y \) and \( -8.7y \) are like terms because they both contain \( y \). Similarly, \( -30 \) and \( 17.4 \) are like terms because they are both constants.
You then combine them by adding or subtracting their coefficients: \( -15y - 8.7y = -23.7y \) and \( -30 + 17.4 = -12.6 \). Combining like terms makes the expression simpler and more manageable.
Algebraic Simplification
Algebraic simplification is the process of reducing an expression to its simplest form. This involves using the distributive property, combining like terms, and performing basic arithmetic operations to make the expression easier to understand or solve.
In our example, the expression \( -7.5(2y + 4) - 2.9(3y - 6) \) was simplified to \( -15y - 30 - 8.7y + 17.4 \) using the distributive property and then further simplified to \( -23.7y - 12.6 \) by combining like terms.
By following these steps: distribute, combine like terms, and then simplify, you can handle even the most complex algebraic expressions effectively.

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Most popular questions from this chapter

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 9 if the sum of its digits is divisible by \(9,\) and not otherwise. Show that (a) \(4,114,107\) is divisible by 9 and (b) \(2,287,321\) is not divisible by \(9 .\)

Write a numerical expression for each phrase and simplify. Twice the product of \(-8\) and \(2,\) subtracted from \(-1\)

Write each sentence as an equation, using \(x\) as the variable. Then find the solution from the set of integers between \(-12\) and \(12,\) inclusive. 7 less than a number is 2

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) $$ (-5+x)(-3+y)(3-a) $$

Simplify each expression. $$ -2(-3 k+2)-(5 k-6)-3 k-5 $$
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