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Chapter 1: Problem 150

Write an equivalent expression using the distributive law. $$ 5 x(y-z+w) $$

Short Answer

Expert verified
\[5xy - 5xz + 5xw\]

Step by step solution

01

Identify the expression

The given expression is: \t\t\[5x(y - z + w)\]
02

Apply the distributive law

The distributive law states that \(a(b + c) = ab + ac\). Apply this law to distribute \t\t5x over each term inside the parentheses (\ty, -z, +w\t): \t\t\[5x \times y - 5x \times z + 5x \times w\]
03

Simplify each term

Multiply \t\t5x\twith each term inside the parentheses:\t\t\t\[\t5xy - 5xz + 5xw\t\]

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They do not have an equal sign like equations. For example, in the expression \(5x(y - z + w)\), \5x\ is being multiplied by another expression in parentheses. Algebraic expressions can be manipulated to make them simpler. This often involves applying various algebraic laws like the distributive property.
In our case, \(5x(y - z + w)\) is an algebraic expression and our goal is to rewrite it in an equivalent form using the distributive law.
Distributive Property
The distributive property is a key concept in algebra that allows us to simplify expressions. It states that for any three numbers \a\, \b\, and \c\, the expression \(a(b + c) = ab + ac\). This means you can distribute, or multiply, \a\ across each term inside the parentheses and then add the results.
For example, if we have the expression \(5x(y - z + w)\), we apply the distributive property by multiplying \5x\ with each term inside the parentheses. So, \(5x \times y - 5x \times z + 5x \times w\).
This action breaks down the original expression into individual terms that are easier to handle.
Simplifying Expressions
Simplifying expressions involves making them as concise and manageable as possible by performing arithmetic operations and combining like terms. Once we have distributed \5x\ across all terms in the original expression \(y - z + w\), we get the new expression \5xy - 5xz + 5xw\.
Each term is simplified by performing the multiplication, resulting in more straightforward algebraic expressions. Simplifying helps in solving algebraic problems more efficiently.
In practice, by simplifying \(5xy - 5xz + 5xw\), we ensure that solving for variables or further manipulations become clearer and less prone to error.

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Most popular questions from this chapter

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?. $$ -8.7,-3,0, \frac{2}{3}, \sqrt{7}, 6 $$

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically. This may involve looking at an earlier exercise or example, or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem. Evaluate each expression using the values provided. $$ (r-s)^{2}-3(2 r-s), \text { for } r=11 \text { and } s=3 $$

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically. This may involve looking at an earlier exercise or example, or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem. Evaluate each expression using the values provided. $$ [3(a-b)]^{2}, \text { for } a=7 \text { and } b=5 $$

The number of animals adopted through Tallahassee Animal Services decreased \(1.1 \%\) from 2011 to \(2012 .\) It then decreased \(3.6 \%\) from 2012 to 2013 and decreased again \(8.5 \%\) from 2013 to \(2014 .\) The number increased \(11.0 \%\) from 2014 to \(2015 .\) If there were 2879 adoptions in \(2015,\) how many were there in \(2011 ?\)

Find an equivalent expression by factoring. $$ 9 p-3 $$
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