1.6 评价表达式

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  评价表达式

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  Introduction
  ::导言

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  Suppose you know the area of a circle is approximately $\begin{array}{r}3.14r^2\end{array}$ , where $\begin{array}{r}r\end{array}$ is the radius of the circle. What if a circle has a radius of 25 inches? How would you find its area? In this section, you'll learn how to substitute 25 inches into the expression , in place of the $\begin{array}{r}r\end{array}$  (radius), and then  evaluate the expression.
  ::假设您知道圆的面积大约是 3. 14r2, 圆的半径是 r 。 如果圆的半径是 25 英寸呢? 您如何找到它的区域 ? 在本节中, 您将学习如何将 25 英寸 替换为表达式, 代替 r( radus) , 然后对表达式进行评估 。

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  Evaluating Expressions
  ::评价表达式

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  In algebra, evaluating an expression commonly means replacing any variables (letters) in the expression with given values, and then simplifying the expression by performing any operations involved. When simplifying, make sure to follow the .
  ::在代数中,评价表达式通常意味着用给定值取代表达式中的任何变量(字母),然后通过执行任何涉及的操作简化表达式。在简化时,请确保跟随 。

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  The following video provides an overview of evaluating expressions, including vocabulary definitions, a general procedure, and examples: 
  ::以下视频概述了评价用语,包括词汇定义、一般程序和实例:

   

   

   

   

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  Examples 
  ::实例

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  Example 1
  ::例1

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  Evaluate $\begin{array}{r}7y-11\end{array}$ , when $\begin{array}{r}y=4\end{array}$ .
  ::评估 7y-11,y=4。

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  Solution:
  ::解决方案 :

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  Evaluate means to follow the directions, which is to take 7 times $\begin{array}{r}y\end{array}$ and subtract 11. Because $\begin{array}{r}y\end{array}$ is the number 4, we can evaluate our expression as follows:
  ::评估方式是遵循方向,即需要7次y和11次减去11次,因为y是数字4,我们可以评估我们的表达方式如下:

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  %20-11%20%26%26%20%5Ctext%7BFirst%2C%20substitute%20the%20number%204%20in%20place%20of%7D%5C%20y.%5C%5C%0A%267%20%5Ctimes%204%20-%2011%20%26%26%20%5Ctext%7BThen%20multiply%7D%5C%207%20%5C%20%5Ctext%7Bby%7D%5C%204.%5C%5C%0A%2628%20-%2011%20%26%26%20%20%5Ctext%7BSubtract%7D%5C%2011%20%5C%20%5Ctext%7Bfrom%7D%5C%2028.%5C%5C%0A%2617%20%26%26%20%5Ctext%7BThe%20solution%20is%7D%5C%2017.">

  $$\begin{array}{rlrl}& 7\times (y)-11& & \text{First, substitute the number 4 in place of}y.\\ & 7\times 4-11& & \text{Then multiply}7\text{by}4.\\ & 28-11& & \text{Subtract}11\text{from}28.\\ & 17& & \text{The solution is}17.\end{array}$$

  
  ::7x- 11 第一, 取代y7x4- 11的编号 4, 然后乘以 7 乘以 4. 28- 11 从 28. 17 乘以 7 乘以 4. 28- 11 。 解决方案为 17 。

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  Because algebra uses variables to represent unknown quantities, the multiplication symbol $\begin{array}{r}\times \end{array}$ is often confused with the variable $\begin{array}{r}x\end{array}$ . To help avoid confusion, mathematicians replace the multiplication symbol with " data-term="Parentheses" role="term" tabindex="0"> parentheses or the multiplication dot •, or by eliminating the symbol entirely if a number is being multiplied by a variable . For example, all four of these expressions have the same meaning:
  ::由于代数使用变量来表示未知的数量,乘数符号 x 往往与变量 x 混为一谈。为了避免混淆,数学家用括号 或乘数点 取代乘数符号,或者如果数字乘以变量,则完全删除符号。例如,所有四个表达式的含义相同:

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  $$\begin{array}{r}4\times a+3\times b\\ 4(a)+3(b)\\ 4\cdot a+3\cdot b\\ 4a+3b\end{array}$$

  
  ::4xa+3xb4(a)+3(b)+4(b)a+3xb4a+3b4a+3bb

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  Example 2
  ::例2

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  Rewrite $\begin{array}{r}P=2\times l+2\times w\end{array}$ with alternative multiplication symbols.
  ::重写 P=2xl+2xw , 另加乘数符号 。

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  Solution:
  ::解决方案 :

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  $\begin{array}{r}P=2\times l+2\times w\end{array}$ can be written as $\begin{array}{r}P=2\cdot l+2\cdot w\end{array}$ .
  ::P=2xl+2xw 可以写成 P=2xl+2xw。

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  It can also be written as $\begin{array}{r}P=2l+2w\end{array}$ .
  ::也可以以P=2l+2w写成。

   

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  The following is a real-life example that shows the importance of evaluating a mathematical expression:
  ::以下是真实生活中的一个实例,表明评价数学表达式的重要性:

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  Example 3
  ::例3

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  To prevent major accidents or injuries, horses are to  be fenced in a rectangular pasture. If the dimensions of the pasture are 300 feet by 225 feet, how much fencing should the ranch hand purchase to enclose the pasture?
  ::为了防止重大事故或伤害,马必须围在长方形草原上,如果牧场面积为300英尺乘225英尺,牧场的手应该购买多少围栏来围住牧场?

  

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  Solution:
  ::解决方案 :

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  Begin by drawing a diagram of the pasture and labeling what you know:
  ::开始绘制牧场图 并标注你所知道的:

  

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  To find the amount of fencing needed, you must add all the sides together—find the perimeter of
  ::为了找到所需的围栏,你必须把两侧加在一起,并找出周围的围栏。

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  the pasture:
  ::牧场:

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  $$\begin{array}{r}P=L+L+W+W=2L+2W\end{array}$$

  
  ::P=L+L+W+W=2L+2W

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  Substituting the values of the variables L and W yields
  ::替代变量L和W的产量值

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  $$\begin{array}{r}P=2\cdot 300+2\cdot 225=1,050\end{array}$$

  
  ::P=2300+2225=1 050

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  feet of fencing.
  ::和栅栏之脚。

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  Example 4
  ::例4

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  a) Write the expression $\begin{array}{r}2\times a\end{array}$ in a more condensed form using alternate multiplication symbols and then evaluate it for $\begin{array}{r}3=a\end{array}$ .
  ::a) 使用其他乘法符号以更压缩的形式写入2xa表达式,然后对3=a进行评价。

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  Solution:
  ::解决方案 :

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  $\begin{array}{r}2\times a\end{array}$ can be written as $\begin{array}{r}2(a),2a,\text{or }2\cdot a.\end{array}$  We can substitute 3 for $\begin{array}{r}a\end{array}$ : 2(3) = 6.
  ::2xa可以写成2(a)、2a或2a。我们可以替换3:2(3)=6。

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  b) If it costs $9.25 for each  movie ticket, how much does it cost for 4 people to see a movie?
  :b) 如果每张电影票价格为9.25美元,4人看电影要花多少钱?

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  Solution:
  ::解决方案 :

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  Since each movie ticket is $9.25, we multiply this price by 4 people to get the total cost:
  ::因为每张电影票是9.25美元,

  $\begin{array}{r}\mathrm{\$}9.25\times 4=\mathrm{\$}37\end{array}$

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  It costs $37 for 4 people to see a movie.
  ::4个人看电影要37美元

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  Example 5
  ::例5

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  Evaluate the following if $\begin{array}{r}a=\text{-}2,c=4,\text{ and }d=\text{-}7.\end{array}$
  ::如果a=2、c=4和d=7,评价以下情况:

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  $$\begin{array}{r}\frac{2a}{c-d}\end{array}$$

   
  
  ::2-d

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  Solution:
  ::解决方案 :

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  Start by substituting in for the variables:
  ::以替换变量开始 :

  $$\begin{array}{r}\frac{2(\text{-}2)}{4-(\text{-}7)}\end{array}$$

   
  

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  Simplify:
  ::简化 :

  $$\begin{array}{r}\frac{2(\text{-}2)}{4-(\text{-}7)}=\frac{\text{-}4}{4+7}=\frac{\text{-}4}{11}\end{array}$$

   
  

   

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  Review  
  ::回顾

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  In 1-4, write the expression in a more condensed form by leaving out the multiplication symbol:
  ::在 1-4 中, 以更压缩的形式写出表达式, 省略乘法符号 :

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  1. $\begin{array}{r}2\times 11x\end{array}$
     ::2x11x 2x11x
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  2. $\begin{array}{r}1.35\cdot y\end{array}$
     ::1.35y
  3. $\begin{array}{r}3\times \frac{1}{4}\end{array}$
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  4. $\begin{array}{r}\frac{1}{4}\cdot z\end{array}$
     ::14z

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  In 5-8, evaluate the expression:
  ::在5-8中,评价以下表述:

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  5. $\begin{array}{r}5m+7\end{array}$ , when $\begin{array}{r}m=3\end{array}$
     ::5m+7,当m=3时
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  6. $\begin{array}{r}\frac{1}{3}(c)\end{array}$ , when $\begin{array}{r}c=63\end{array}$
     ::13(c),c=63时
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  7. $\begin{array}{r}(k-11)÷8\end{array}$ , when $\begin{array}{r}k=43\end{array}$
     :k=43) k=43,8,k=43
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  8. $\begin{array}{r}(-2{)}^2+3(j)\end{array}$ , when $\begin{array}{r}j=-3\end{array}$
     :-2)2+3(j)

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  In 9-13, evaluate the expression. Let $\begin{array}{r}a=\text{-}3,b=2,c=5,\text{ and }d=\text{-}4.\end{array}$
  ::9-13,评价表达式。让我们a=3,b=2,c=5,d=4。

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  9. $\begin{array}{r}4c+d\end{array}$
     ::4c+d 4c+d
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  10. $\begin{array}{r}5ac-2b\end{array}$
      ::5ac-2b
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  11. $\begin{array}{r}\frac{3b}{d}\end{array}$
      ::3bd 3bd
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  12. $\begin{array}{r}\frac{a-4b}{3c+2d}\end{array}$
      ::a-4b3c+2d
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  13. $\begin{array}{r}\frac{ab}{cd}\end{array}$
      ::abcd abcd

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  In 14-18, evaluate the expression. Let $\begin{array}{r}x=\text{-}1,z=\text{-}3,\text{ and }w=4.\end{array}$
  ::在14-18中,评价表达式。让 x=-1, z=-3, 和 w=4。

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  14. $\begin{array}{r}8x^3\end{array}$
      ::8x3
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  15. $\begin{array}{r}3z^2-5w^2\end{array}$
      ::3z2-5w2
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  16. $\begin{array}{r}\frac{z^3+w^3}{z^3-w^3}\end{array}$
      ::z3+w3z3-w3 z3 -w3
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  17. $\begin{array}{r}2x^2-3x^2+5x-4\end{array}$
      ::2x2-3x2+5x-4
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  18. $\begin{array}{r}3+\frac{1}{z^2}\end{array}$
      ::3+1z2 3+1z2

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  In 19-23, evaluate the expression in each real-life problem:
  ::19-23年,评价每个现实生活中问题的表达方式:

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  19. The measurement around the widest part of these holiday bulbs is called their circumference. The formula for circumference is $\begin{array}{r}2(r)\pi \end{array}$ , where $\begin{array}{r}\pi \approx 3.14\end{array}$ and $\begin{array}{r}r\end{array}$ is the radius of the circle. Suppose the radius is 1.25 inches. Find the circumference .  

      

      
      ::围绕这些假日灯泡最宽部分的测量称为环绕。 环绕的公式是 2(r), 其中 3. 14 和 r 是圆的半径。 假设半径为 1. 25 英寸。 找到环绕 。
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  20. The dimensions of a piece of notebook paper are 8.5 inches by 11 inches. Evaluate the area of the paper. The formula for the area of a rectangle is length $\begin{array}{r}\times \end{array}$ width.
      ::笔记本纸的尺寸为 8.5 英寸乘11 英寸。 评估纸的面积。 矩形区域的公式是长度 × 宽度 。
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  21. Sonya purchased 16 cans of soda at $0.99 each. What is the amount Sonya spent on soda?
      ::Sonya每人以0.99美元购买了16罐苏打水。Sonya在苏打水上花费的金额是多少?
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  22. Mia works at a job earning $4.75 per hour. How many hours should she work to earn $124?
      ::米娅工作的工作每小时收入4.75美元,她要工作多少小时才能挣124美元?
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  23. The area of a square is the side length squared. Evaluate the area of a square with a side length of 10.5 miles.
      ::方形区域为侧长方形区域。以侧长10.5英里对方形区域进行评估。

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  Review (Answers)
  ::回顾(答复)

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  Please see the Appendix. 
  ::请参看附录。