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Chapter 5: Problem 10

Simplify by clearing parentheses and combining like terms. $$-3(9.2 w-4.1)+3.62 w$$

Short Answer

Expert verified
-23.98w + 12.3

Step by step solution

01

Distribute -3 to terms inside the parentheses

First, apply the distributive property to \footnote -3\footnote inside the parentheses. Multiply each term inside the parentheses by -3\footnote: \[ -3(9.2w - 4.1) = -3 \times 9.2w + (-3) \times (-4.1) \] This simplifies to: \[ -27.6w + 12.3 \]
02

Combine like terms

Add the term containing :w\footnote from the distribution to 3.62w: \[ -27.6w + 3.62w + 12.3 \] Now combine like terms: \[ (-27.6 + 3.62)w + 12.3 \] Simplify the coefficient of :w\footnote: \[ -23.98w + 12.3 \]

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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 key concept in algebra that allows you to simplify expressions by distributing a single term across the terms inside parentheses. This can make complex problems easier to handle.

For example, with the expression \(-3(9.2w - 4.1)\), the -3 is distributed across both terms inside the parentheses. You do this by multiplying -3 by each term:
\(-3 \times 9.2w\) and \(-3 \times -4.1\).

Here's how it looks:
\(-3 \times 9.2w = -27.6w\) and \(-3 \times -4.1 = 12.3\).

So, \(-3(9.2w - 4.1)\) simplifies to \(-27.6w + 12.3\).

This first step sets the stage for further simplification by making the expression easier to work with.
Combining Like Terms
After using 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. In our example, like terms are those that contain the variable value \w\.

Once we have the terms \(-27.6w\) and \(3.62w \) in the expression, we can combine them:
\(-27.6w + 3.62w\).

To combine them, you simply add or subtract their coefficients (the numerical values in front of the variables). In this case:
\(-27.6 + 3.62\) equals \(-23.98\).

Therefore, \(-27.6w + 3.62w \) simplifies to \(-23.98w\).

Make sure to combine only like terms together. Mixing different terms can lead to incorrect results.
Coefficients
Coefficients are the numerical parts of the terms in algebraic expressions. They are important because they tell you how many times to multiply the variable.

For example, in the term \(-27.6w\), the coefficient is \(-27.6\). In the term \(3.62w \), the coefficient is \(3.62\).

When combining like terms, you deal primarily with their coefficients. Adding the coefficients gives you a single term with a new coefficient, simplifying the expression:
\(-27.6 + 3.62 = -23.98\).

Therefore, the final expression becomes \(-23.98w\).

Understanding coefficients helps you simplify expressions effectively and solve equations more easily.

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Most popular questions from this chapter

Six test scores for Jonathan's history class are listed. Find the mean and median. Round to the nearest tenth if necessary. Did the mean or median give a better overall score for Jonathan's performance? $$92 \%, 98 \%, 43 \%, 98 \%, 97 \%, 85 \%$$

Write the fractions as decimals. Explain how to memorize the decimal forms for these fractions with denominator of 3 a. \(\frac{1}{3} \quad\) b. \(\frac{2}{3}\)

For Exercises 67鈥�75, divide. Round the answer to the indicated place value. Use the rounded quotient to check. $$1.8 \div7 \text { hundredths }$$

For Exercises \(27-32,\) subtract the decimal numbers. Then round the numbers and find the difference to determine if your answer is reasonable. The first estimate is done for you. $$53.9-22.4$$

For Exercises \(33-40\), convert the fraction to a decimal and round to the indicated place value. \(-\frac{1}{8} ;\) hundredths
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