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Chapter 5: Problem 18

Use the distributive property to rewrite each expression without using parentheses. a. \(-2(x+8)\) b. \(4(0.75-y)\) c. \(-(z-5)\)

Short Answer

Expert verified
a. \\(-2x - 16\\); b. \\(3 - 4y\\); c. \\(-z + 5\\).

Step by step solution

01

Distribute Within the Expression a.

For the expression \(-2(x + 8)\), apply the distributive property: multiply \-2\ by each term inside the parentheses. Calculate \-2 \times x = -2x\ and \-2 \times 8 = -16\.
02

Rewrite Expression a Without Parentheses.

Combine the results of the distribution in Step 1 to get the expression without parentheses: \(-2(x + 8) = -2x - 16\).
03

Distribute Within the Expression b.

For the expression \(4(0.75 - y)\), apply the distributive property: multiply \4\ by each term inside the parentheses. Calculate \4 \times 0.75 = 3\ and \4 \times (-y) = -4y\.
04

Rewrite Expression b Without Parentheses.

Combine the results from Step 3 to get the expression without parentheses: \(4(0.75 - y) = 3 - 4y\).
05

Distribute Within the Expression c.

For the expression \(-(z - 5)\), apply the distributive property: multiply \-1\ by each term inside the parentheses. Calculate \-1 \times z = -z\ and \-1 \times (-5) = +5\.
06

Rewrite Expression c Without Parentheses.

Combine the results from Step 5 to get the expression without parentheses: \(-(z - 5) = -z + 5\).

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Key Concepts

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

Algebraic Expressions
Understanding algebraic expressions is a foundational skill in mathematics. An algebraic expression is a mathematical phrase that can include numbers, variables, and operators like addition, subtraction, and multiplication. In expressions like \(-2(x+8)\), \(4(0.75-y)\), and \(-(z-5)\), the goal is to simplify them using various mathematical rules. These expressions can be thought of as recipes involving different components:
  • **Numbers** are the constants, such as \(-2\), \(4\), \(0.75\), and \(-1\) (often implied).
  • **Variables** are symbols like \(x\), \(y\), and \(z\) that represent unknown values.
  • **Operators** tell us how to relate these components, indicating calculations like multiplying and adding.
As soon as you understand how to interpret the parts of these expressions, you are better equipped to apply mathematical principles like the distributive property for simplifying.
Parentheses Removal
Removing parentheses in algebraic expressions is essential when simplifying equations or expressions. The primary tool for this task is the distributive property, which unfolds the compounded operations inside parentheses sections. Take the expression \(-2(x + 8)\) as an example:
  • **Identify Terms Inside Parentheses:** The terms inside the parentheses are \(x\) and \(8\).
  • **Apply the Distributive Property:** Multiply \(-2\) by each term inside. This means \(-2 \times x\) and \(-2 \times 8\).
  • **Rewrite Without Parentheses:** This results in a simpler expression, \(-2x - 16\).
Repeat this process with other expressions, such as \(4(0.75 - y)\), processing each term separately and leaving a simplified algebraic expression that鈥檚 free of parentheses. It's like opening a mathematical box to see and adjust its contents individually.
Mathematical Operations
Mathematical operations are the actions we take to manipulate algebraic expressions. Key operations include multiplication and addition, both crucial in the process of parentheses removal using the distributive property. Let's delve deeper:
  • **Multiplication:** Essential in applying the distributive property, this operation distributes an external multiplier across terms inside parentheses. For instance, in \(-(z - 5)\), the \(-1\) multiplies with each term inside, converting everything inside to opposite signs.
  • **Addition/Subtraction:** Used to combine or subtract terms after distribution, resulting in a streamlined expression without parentheses, such as \(-z + 5\) instead of \(-(z-5)\).
Each operation follows specific rules, ensuring the mathematical integrity of expressions. Being adept in these operations will enhance your algebraic simplification skills significantly.

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Most popular questions from this chapter

Mrs. Abdul mixes bottled fruit juice with natural orange soda to make fruit punch for a party. The bottled fruit juice is \(65 \%\) real juice and the natural orange soda is \(5 \%\) real juice. How many liters of each are combined to make 10 liters of punch that is \(33 \%\) real juice?

APPLICATION Will is baking a new kind of bread. He has two different kinds of flour. Flour \(\mathrm{X}\) is enriched with \(0.12 \mathrm{mg}\) of calcium per gram; Flour \(\mathrm{Y}\) is enriched with \(0.04 \mathrm{mg}\) of calcium per gram. Each loaf has \(300 \mathrm{~g}\) of flour, and Will wants each loaf to have \(30 \mathrm{mg}\) of calcium. How much of each type of flour should he use for each loaf? a. Will wrote this system of equations: $$ \left\\{\begin{aligned} x+y &=300 \\ 0.12 x+0.04 y &=30 \end{aligned}\right. $$ Give a real-world meaning to the variables \(x\) and \(y\), and describe the meaning of each equation. b. Write a matrix for the system. c. Find the solution matrix. d. Explain the real-world meaning of the solution.

Write an equation in point-slope form using the given information. a. A line that passes through the point \((5,-3)\) and has slope \(-2\). b. A line that passes through the point \((-3,7)\) and has slope \(2.5\).

Substitute \(4-3 x\) for \(y\). Then rewrite each expression in terms of one variable. a. \(5 x+2 y\) b. \(7 x-2 y\) (a)

At the Coffee Stop, you can buy a mug for \(\$ 25\) and then pay only \(\$ 0.75\) per hot drink. a. What is the slope of the equation that models the total cost of refills? What is the real-world meaning of the slope? b. Use the point \((33,49.75)\) to write an equation in point-slope form that models this situation. c. Rewrite your equation in intercept form. What is the real-world meaning of the \(y\)-intercept?
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