将文字转换为代数

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In your daily life (except in math class), no one asks you to "solve  $\begin{align*}4x+35=115\end{align*}$ ." However, your boss might ask you to determine how much you can spend on snacks for 4 meetings at a bulk grocery store with a membership of $35 if the budget is $115. You can express this as an algebraic statement and solve for the unknown value. In this section, we see how to do this.  
::在你的日常生活中(数学课除外),没有人要求你“解决 4x+35=115 ” 。然而,你的上司可能会要求你确定,如果预算为115美元,那么,你可以用这个代数语句来表达这一点,并解决未知值。在本节中,我们可以看到如何做到这一点。

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Translating Verbal Expressions Into Algebraic Expressions 
::将文字表达式转换成代数表达式

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When translating a verbal expression , you first need to identify the unknown and assign it a variable . It is often helpful to write this down. For example, in the scenario from the introduction, the unknown is the amount that can be spent for snacks for each meeting. We can write
::当翻译口头表达式时,您首先需要识别未知的表达式, 并指定一个变量。 写下来通常有用 。 例如, 在引言的假想中, 未知的是每次会议可以用于零食的金额 。 我们可以写入

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$$\begin{align*}m=\ \text{the amount of money for snacks at a meeting}\end{align*}$$
::m=会议零食的金额

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This is called a legend and you can always refer back to it when you are translating your expression . Now that we have defined our unknown, we need to identify words that represent operations . 
::这被称为传说, 当您翻译您的表达方式时, 您总是可以引用它。 既然我们已经定义了未知, 我们需要识别代表操作的单词 。

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Addition
::加加加

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The following words in a verbal expression represent addition .
::口头表达中的以下词语是补充。

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Example 1
::例1

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Abdul has $5 more than Xavier has. Write an algebraic expression to show the number of dollars Abdul has.
::Abdul比Xavier多出5美元。 写一个代数表达式来显示 Abdul 的美元数 。

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Solution:  Let  $\begin{align*}d\end{align*}$  be the amount of money Xavier has. Since more than represents addition, we can write  $\begin{align*}d+5\end{align*}$  to represent the amount of money Abdul has.
::解决方案 : 让Xavier拥有的金额成为解决方案 。 由于超过附加值, 我们可以写入 d+5 来代表 Abdul 拥有的金额 。

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Subtraction
::减法减法

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The following words in a verbal expression represent subtraction .
::以口头表示的以下词语表示减法。

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Notice that for some of the phrases above, like 7 subtracted from x, the order that the words appear in is opposite the order that we translate the words into algebra.
::请注意,对于上文的一些短语,如7个从 x 中减去,单词出现的顺序与我们将单词转换为代数的顺序相反。

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Example 2
::例2

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Lian's height is  $\begin{align*}b\end{align*}$  inches less than Hannah's height. Hannah is 65 inches tall. Write an algebraic expression to show Lian’s height in inches.
::Lian的身高比Hannah的身高低2英寸。 Hannah身高65英寸。写代数表达法来显示Lian的身高。

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Solution:  We are given a variable for our unknown. Being careful of order, we put 65 first and get $\begin{align*}65-b\end{align*}$  for Lian's height. 
::解答: 我们有一个未知变量。 注意秩序, 我们先放65个, 再取65个到Lian的高度。

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Multiplication 
::乘法乘法

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The following words in a verbal expression represent multiplication .
::以口头表示的以下词语表示乘数。

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Example 3
::例3

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Colleen spends $12 for each ticket. Write an algebraic expression to show how much Colleen spends on tickets.
::科琳每张票要花12美元 写代数表达法来显示科琳在票上花多少钱

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Solution: Let the cost of each ticket be t . Therefore , Colleen spends 12t on tickets.
::解决:让每张票的成本为t。 因此,Colleen在票上花费了12吨。

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Division
::司司司

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The following words in a verbal expression represent division .
::口头表达的以下文字表示分裂。

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Example 4
::例4

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The professor divided the class of 30 students into $\begin{align*}n\end{align*}$  equal groups. Write an algebraic expression to show how many students are in each group.
::教授将30名学生分成n等类。写一个代数表达式,显示每个组有多少学生。

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Solution:  A variable is already defined for the unknown, so we recognize the operation "divided into" and write  $\begin{align*}\frac{30}{n}\end{align*}$  for the number of students in each group. 
::解决方案:一个变量已经定义了未知的变量, 所以我们识别“ 分裂成” 操作, 并写下每个组的学生人数 30n 。

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Translating Verbal Statements Into Algebraic Statements
::将口头发言译成代数发言

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In this chapter, we have discussed two types of statements—equalities and inequalities. Let's see the words we can recognize to translate these types of verbal statements into algebraic statements.
::在本章中,我们讨论了两种声明——不平等和不平等,让我们看看我们能够承认的词语,将这些类型的口头声明变成代数声明。

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The following words in a verbal description represent an equal sign.
::口头描述中的以下词语代表一个相同的符号。

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*Note that any of the tenses and numbers of the verb "to be" work in this case. For example, "are," "were," "will be," etc.
::* 注意在此情况下任何动词“ 将” 的时态和数字。 例如, “ 是 ” , “ were” , “ 将是 ” 等 。

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The following words in a verbal description represent an inequality .
::口头描述中的以下词语代表一种不平等。

Ways to represent  $\begin{align*}x < 4\end{align*}$	Ways to represent  $\begin{align*}x>4\end{align*}$
x is less than 4	x is greater than 4
4 is more than x	4 is less than x

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Note that the difference between these expressions and those of addition and subtraction is the word "is." Any tense of the verb "to be" could be used instead of "is." 
::请注意,这些表达式与添加和减法的表达式之间的区别是“是”一词。动词“待发”的任何时态都可以代替“是”。

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Example  5
::例5

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Elise is taking pilates  classes. The first class is twice as expensive as each additional class. She spends $270 for 8 classes. How much was the first class?
::伊莉斯正在上小费班。 头等舱的费用是每增加一班的两倍。 她花270美元用于8班。 头等舱多少钱 ?

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Solution:  Translate each statement.
::解答:翻译每个语句。

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Call the regularly priced classes  $\begin{align*}l\end{align*}$ . Then, the first class will be twice as expensive, or  $\begin{align*}2l\end{align*}$ . Together, the classes cost  $\begin{align*}2l+7l\end{align*}$ ,  $\begin{align*}2l\end{align*}$  for the first class and  $\begin{align*}7l\end{align*}$  for the remaining 7 classes.
::然后,头等舱价格将是贵价的两倍,或者2升。 一起,头等舱成本为2升+7升,头等舱成本为2升,头等舱成本为2升,其余7级成本为7升。

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"She spends $270 for 8 classes" or the cost of the class is $270:  $\begin{align*}2l + 7l = \$ 270\end{align*}$
::"她花270元8个班级" 或者这个班的费用是270美元 2l+7l=270美元

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Now, we have an algebraic statement to solve:
::现在,我们有一个代数声明要解决:

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$$\begin{align*}2l+7l &= 270\\ 9l &= 270\\ l &= 30\end{align*}$$
::2l+7l=270l=30

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The regularly priced classe s are $30. The first class will be $60.
::定期定价的班级为30美元,头等舱为60美元。

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Example 6
::例6

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A town with a population of 9,000 people declines by 45 people on average each year. When will the population of the town be less than 8,000 people?
::一个人口为9 000人的城镇每年平均减少45人,人口将少于8 000人。

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Solution:  Let  $\begin{align*}y\end{align*}$  be the number of years. Then, the population declines by 45 people each year or  $\begin{align*}45y\end{align*}$ .
::解决方案:让年数成为问题。然后,人口每年减少45人,即45人。

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Declines means the number reduces by, so we need to subtract: $\begin{align*}9,000-45y\end{align*}$ . This is the population of the town.
::下降意味着数量减少,所以我们需要减去:9,000-45y。这是该镇的人口。

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We want to determine when the population of the town will be less than  8,000 or $\begin{align*}9,000-45y<8,000\end{align*}$ .
::我们想确定该镇的人口将少于8,000人或9,000-45y<8,000人。

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We can now solve this inequality
::我们现在可以解决这种不平等了

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$$\begin{align*}\cancel{9,000}-45y<8,000 \\ \underline{-\cancel{9,000} \qquad \ -9,000} \\ \frac{-\cancel{45}y}{-\cancel{45}} < \frac{-1,000}{-45} \\ y>22.22\end{align*}$$
::9,000-45y < 8,000-9,000-9,000-9,000- -45-45-1,000-45y>22。

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The town will have a population of less than 8,000 people in 23 years.  
::23年内,该镇人口将少于8 000人。

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Example 7 
::例7

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The revenue of a large multinational corporation exceeds 649 times the revenue of a small business. If the revenue of a small business is $3.6 million, then what is the the revenue of the large corporation? 
::大跨国公司的收入是小企业收入的649倍,如果小企业的收入是360万美元,那么大公司的收入是什么?

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Solution:  Let  $\begin{align*}R\end{align*}$  be the revenue of the large corporation. 
::解决办法:让R成为大公司的收入。

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We are given that  $\begin{align*}R\end{align*}$  exceeds or is greater than  649 times the revenue of a small business. Since the revenue of the small business is $3,600,000, we have  $\begin{align*}R > 649 \times 3,600,000\end{align*}$  or  $\begin{align*}R > 2,336,400,000\end{align*}$ . The revenue exceeds $2.3364 billion.
::我们了解到R超过或超过小企业收入的649倍,由于小企业收入为3 600 000美元,我们有649×3 600 000卢比或2 336 400 000卢比,收入超过23.364亿美元。

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Summary
::摘要

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• Identify the unknown in the expression or statement by assigning it a variable.
  ::指定一个变量来识别表达式或语句中的未知值。
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• Look for key words in the verbal statements to translate into algebra.
  ::寻找口头发言中的关键词,将其翻译成代数。

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Review
::回顾

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Translate each of the statements into algebra and solve. 
::将每个语句转换为代数并解析 。

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1. The sum of two consecutive numbers is 79. Find the two numbers.
   ::连续两个数字的总和是79 找出这两个数字。
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2. The sum of two consecutive odd numbers is 44. Find the two odd numbers.
   ::连续两个奇数的总和是44 找出这两个奇数
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3. The sum of three consecutive even numbers is 138. What are the three numbers?
   ::连续三个偶数的总和是138 这三个数字是多少?
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4. Harriet bakes and sells cookies to local stores. Her cost for one dozen cookies is $2.75 and she sells them to stores for $7.00 (per dozen). How many dozen cookies does she need to make to earn $500? Round to the nearest dozen.
   ::哈丽特烤面包和向当地商店出售饼干。她十几个饼干的成本是2.75美元,卖给商店(每12美元)7美元。她要挣500美元,她需要挣多少个饼干?圆到最近的12美元。
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5. A football field is a rectangle where the length is 120 yards. If the total perimeter is 1040 feet, what is the width of a football field? Leave your answer in feet.
   ::足球场是一个长120码的矩形。 如果总周界是1040英尺,那么足球场的宽度是多少? 请不要回答。
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6. George is building a rectangular, fenced-in dog run. He has 120 feet of fencing and wants the length to be 20 feet greater than the width. If you use all the fencing, find the length and width of the dog run.
   ::George正在建造一条长方形、有栅栏的狗跑。他有120英尺的栅栏,希望长度大于宽度20英尺。如果你使用所有的栅栏,请找到狗跑的长度和宽度。
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7. The average speed on a highway is 65 miles per hour (mph). Assuming you drive the speed limit, how long will it take you to drive 350 miles? Use the formula $\begin{align*}distance = rate \cdot time\end{align*}$ . Round your answer to two decimal places.
   ::高速公路上的平均速度是每小时65英里( mph ) 。 假设您驾驶速度限制, 您需要多久才能驾驶350英里? 使用公式的距离=时间。 您的回答四舍五入到小数点后两个位数 。
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8. Using the information in #7, how many miles did you drive on the highway if you drove for 2.5 hours?
   ::使用7号公路的信息 如果你开2.5小时的话 你开高速公路开多少英里?
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9. If the cost of a pair of sneakers is $49.99 and the sales tax is 5.5%, can Dominic afford the sneakers if he has $55?
   ::如果一双运动鞋的费用是49.99美元,而销售税是5.5%,那么如果多米尼克有55美元,他能买得起运动鞋吗?
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10. A store's cost for a tablet is $250 and the store sells the tablet for a sticker price of $450. By what percent of the sticker price can the store sell the tablet for to exceed the cost of the tablet? 
    ::平板电脑的商店成本是250美元,商店出售平板电脑的贴纸价格是450美元。 以贴纸价格的多少百分比的价格,商店出售平板电脑的成本能超过平板电脑的成本?

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Answers for Review Problems
::回顾问题的答复

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Please see the Appendix.
::请参看附录。