课程： 12.7变量表达式的积与商

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选择节变量表达式的积与商

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变量表达式的积与商
Gloria’s mom, Mrs. Thomas, has given Gloria permission to have a sleepover. Gloria has invited 8 of her friends to the sleepover. Mrs. Thomas is going to pick up pizzas and hot wings for the 9 girls to feast on while they watch a movie. Ken, the guy at the pizzeria, tells Mrs. Thomas that a large pizza contains 12 slices. Mrs. Thomas wants each of the girls to have at least 4 slices, so she needs a total of 36 slices. How can she use this information to write a variable expression that will help her decide how many pizzas she should pick up?
::Gloria的妈妈Thomas女士允许Gloria去过夜。Gloria邀请了8位朋友到过夜。Thomas夫人准备在9名女孩看电影时去取披萨和热翼,让9名女孩一起吃。披萨店的人Ken告诉Thomas夫人,大披萨里有12片。Thomas夫人希望每个女孩至少有4片,所以她需要总共36片。她如何利用这些信息写一个可变表达方式来帮助她决定应该拿多少披萨呢?

In this concept, you will learn to write and simplify products and quotients of variable expressions.
::在此概念中,您将学会写作和简化各种表达式的产品和商数。

Writing Products and Quotients of Variable Expressions
::变量表达式的书写产品和引号

To evaluate an expression you must have a given value for the variable. If you have not been given a value for the variable, then you simplify the expression.
::要评价一个表达式,您必须拥有变量的给定值。如果您没有给变量给定值,那么您将简化表达式。

Here is an expression.
::这是一种表达方式。

$\begin{aligned} 3 (5 x) \end{aligned}$

This is a multiplication problem. The number next to the " data-term="Parentheses" role="term" tabindex="0"> parentheses means that you are going to multiply. However, you haven’t been given a value for $\begin{aligned} x \end{aligned}$ so you are going to simplify this expression. You can’t do anything with the variable, but you can multiply the numerical part of each term .
::这是一个乘法问题。括号旁边的数字意味着您要乘数。但是, 您还没有给 x 给定值, 所以您会简化这个表达式。您不能对变量做任何事情, 但是您可以将每个术语的数值部分乘以。

$\begin{aligned} \begin{array}{rcl} 3 (5 x) \\ 3 \times 5 = 15 \end{array} \end{aligned}$

Keeping the ‘ $\begin{aligned} x \end{aligned}$ ’ variable, the expression is simplified as $\begin{aligned} 15 x \end{aligned}$ .
::保留“ x” 变量时,表达式简化为 15 x 。

The answer is $\begin{aligned} 15 x \end{aligned}$ .
::答案是15x。

Let’s look at another example.
::让我们再看看另一个例子。

$\begin{aligned} 4 \cdot (12 \cdot y) \end{aligned}$

These dots mean to multiply.
::这些点意味着成倍增加。

First, complete the operation in parentheses.
::首先,在括号中填写完成操作。

$\begin{aligned} 12 times y = 12 y \end{aligned}$

Now, the expression becomes $\begin{aligned} 4 \cdot (12 y) \end{aligned}$ .
::现在,表达式变成 4 ( 12 y) 。

Next, multiply the number parts of the two terms .
::接下来,乘以这两个术语的编号部分。

$\begin{aligned} 4 \times 12 = 48 \end{aligned}$

Then, keeping the ‘ $\begin{aligned} y \end{aligned}$ ’ variable, the expression is simplified as $\begin{aligned} 48 y \end{aligned}$ .
::然后,保留“y”变量,表达式简化为48 y。

The answer is $\begin{aligned} 48 y \end{aligned}$ .
::答案是48 y 。

You can also simplify quotients. Remember that a quotient means division .
::您也可以简化商数。记住, 商数意味着分割。

Here is another expression.
::下面是另一种表达方式。

$\begin{aligned} \frac{4 x}{2 x} \end{aligned}$

Whether it is a number or a variable, a fraction over itself is equal to one.
::无论是一个数字还是一个变量,一个分数本身等于一个。

$\begin{aligned} \frac{6}{6} = 1 \end{aligned}$

This is called simplifying a fraction. Variables can also be simplified when you divide. When simplifying a quotient with a variable expression, simplify the variables first.
::这被称为简化一个分数。变量也可以在分隔时被简化。当用变量表达式简化一个商数时, 请先简化变量。

For example:
::例如:

$\begin{aligned} \frac{4 x}{2 x} = \frac{4}{2} \end{aligned}$

Remember, $\begin{aligned} x \end{aligned}$ divided by $\begin{aligned} x \end{aligned}$ is equal to one. So, the $\begin{aligned} x \end{aligned}$ ’s cancel each other out.
::记住, x 除以 x 等于 1。因此, x 互相取消。

Then, you are left with four divided by two.
::然后,你们将四人除以二。

$\begin{aligned} 4 \div 2 = 2 \end{aligned}$

The answer is 2.
::答案是2。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the slumber party feast.
::早些时候,有人给了你一个关于睡衣派对的麻烦

Each pizza has 12 slices and Gloria’s mom wants each of the nine girls to get at least four slices, which means she needs a total of at least 36 slices. How can she use this information to write a variable expression that will help her decide how many pizzas she should pick up?
::每个披萨都有12片,而Gloria的妈妈希望9个女孩中的每个都能得到至少4片,这意味着她需要总共至少36片。她如何利用这些信息来写一个可变表达方式来帮助她决定她应该拿多少披萨呢?

Begin by expressing the situation in a phrase.
::首先用一句话来表达情况。

She needs ‘ $\begin{aligned} x \end{aligned}$ ’ number of pizzas each with 12 slices to be divided by 9 girls.
::她需要 " x " 份的披萨数量,每份12片,除以9个女孩。

First, identify the key words, variables, and numbers.
::首先,确定关键词、变量和数字。

12 is the number of slices in each pizza
::12是每个比萨饼的切片数

$\begin{aligned} x \end{aligned}$ is the unknown number of pizzas
::x 是数量不详的比萨饼数量

“each with” suggests multiplication
::“每一与” 表示乘法

“divided by” suggests division
::“以”建议“除以”表示

9 is the number of girls
::9是女孩人数

Next, write the phrase as an expression.
::下一位, 将短语写成表达式。

‘ $\begin{aligned} x \end{aligned}$ ’ number of pizzas each with 12 slices becomes $\begin{aligned} 12 (x) \end{aligned}$
::" x " 每个12片的披萨数量变为12(x)

$\begin{aligned} 12 x \end{aligned}$

divided by 9 girls becomes $\begin{aligned} \frac{◻}{9} \end{aligned}$
::女孩除以9, 女孩成为 9

$\begin{aligned} \frac{◻}{9} \end{aligned}$

Then, put it together as an expression.
::然后,把它拼起来作为一种表达方式。

$\begin{aligned} \frac{12 x}{9} \end{aligned}$

Note that ‘ $\begin{aligned} x \end{aligned}$ ’ is the only variable in the expression, so you can’t do anything with the $\begin{aligned} x \end{aligned}$ .
::请注意,`x ' 是表达式中唯一的变量,所以您不能对 x 做任何事情。

Now, simplify the number values by dividing 12 and 9 by their greatest common factor , 3.
::现在,简化数字值,将12和9除以其最大的共同系数3。

$\begin{aligned} \begin{array}{rcl} 12 \div 3 & = & 4 \\ 9 \div 3 & = & 3 \end{array} \end{aligned}$

This becomes $\begin{aligned} \frac{4}{3} \end{aligned}$ .
::这变成了4 3。

Finally, add the $\begin{aligned} x \end{aligned}$ in.
::最后,加上x英寸。

$\begin{aligned} \frac{4 x}{3} \end{aligned}$

The answer is $\begin{aligned} \frac{4 x}{3} \end{aligned}$ .
::答案是 4 x 3 。

Example 2
::例2

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} \frac{8 x y}{4 x} \end{aligned}$

There are like $\begin{aligned} x \end{aligned}$ ’s in both the numerator and denominator.
::分子和分母中都有类似x的分母。

First, simplify the $\begin{aligned} x \end{aligned}$ ’s to 1 because any number or variable divided by itself is equal to 1. Here is what you are left with.
::首先,将 x 简化为 1, 因为任何数字或变量本身除以等于 1 。这是您剩下的部分。

$\begin{aligned} \begin{array}{rcl} \frac{8 (1) y}{4 (1)} \\ \frac{8 y}{4} \end{array} \end{aligned}$

Next, simplify the number values by dividing four into eight.
::接下来,简化数字值,将四分之八。

$\begin{aligned} 8 \div 4 = 2 \end{aligned}$

Then, replace the $\begin{aligned} y \end{aligned}$ .
::然后,替换Y。

The answer is $\begin{aligned} 2 y \end{aligned}$ .
::答案是 2 Y 。

Example 3
::例3

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} 5 \cdot (4 \cdot x) \end{aligned}$

Remember, the dots mean to multiply.
::记住,点意味着乘以。

First, complete the operation in parentheses.
::首先,在括号中填写完成操作。

$\begin{aligned} 4 times x = 4 x \end{aligned}$

The expression becomes $\begin{aligned} 5 \cdot (4 x) \end{aligned}$ .
::表达式变为 5 ( 4 x ) 。

Next, multiply the number parts of the two terms.
::接下来,乘以这两个术语的编号部分。

$\begin{aligned} 5 \times 4 = 20 \end{aligned}$

Then, keeping the ‘ $\begin{aligned} x \end{aligned}$ ’ variable, the expression is simplified as $\begin{aligned} 20 x \end{aligned}$ .
::然后,保留 ' x ' 变量,表达式简化为 20 x 。

The answer is $\begin{aligned} 20 x \end{aligned}$ .
::答案是20x。

Example 4
::例4

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} \frac{6 y}{3} \end{aligned}$

First, simplify the number values by dividing 6 by 3.
::首先,简化数字值,将6除以3。

$\begin{aligned} 6 \div 3 = 2 \end{aligned}$

The answer is $\begin{aligned} 2 y \end{aligned}$ .
::答案是 2 Y 。

Example 5
::例5

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} \frac{14 x y}{7 y} \end{aligned}$

First, simplify the like $\begin{aligned} y \end{aligned}$ ’s in the numerator and denominator to 1 and cancel them out. Here is what you are left with.
::首先,简化您在分子和分母中的位置,然后取消它们。这里是剩下的。

$\begin{aligned} \begin{array}{rcl} \frac{14 x (1)}{7 (1)} \\ \frac{14 x}{7} \end{array} \end{aligned}$

Next, simplify the number values by dividing 7 into 14.
::下一步,简化数字值,将 7 分成 14 。

$\begin{aligned} 14 \div 7 = 2 \end{aligned}$

The answer is $\begin{aligned} 2 x \end{aligned}$ .

::答案是 2 x 。

Review
::回顾

Simplify the following expressions.
::简化下列表达式。

$\begin{aligned} 5 x (4) \end{aligned}$
::5 x (4 )

$\begin{aligned} 3 (5 x) \end{aligned}$
::3 (5x)

$\begin{aligned} 4 y (2) \end{aligned}$
::4 y ( 2 )

$\begin{aligned} 2 \cdot (8 \cdot a) \end{aligned}$
::2______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} 4 \cdot (7 x) \end{aligned}$
::4 ( 7 x )

$\begin{aligned} \frac{9 y}{3 y} \end{aligned}$
::9 y 3 y 年 9 y 3 y

$\begin{aligned} \frac{64 x}{8 x} \end{aligned}$
::64 x 8 x 8 x x

$\begin{aligned} \frac{12 x y}{4} \end{aligned}$
::12 x y 4 12 x y 4

$\begin{aligned} \frac{10 a b}{5 a} \end{aligned}$
::10 a b 5 a

$\begin{aligned} \frac{18 x y}{6 y} \end{aligned}$
::18 x y 6 y

$\begin{aligned} 15 x (4 y) \end{aligned}$
::15 x( 4 y )

$\begin{aligned} 7 x (2) \end{aligned}$
::7 x ( 2 )

$\begin{aligned} 12 x (4 y) \end{aligned}$
::12 x( 4 y )

$\begin{aligned} 13 x (4 y) \end{aligned}$
::13 x( 4 y )

$\begin{aligned} 12 x (7) \end{aligned}$
::12 x (7 )

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Resources
::资源

章节大纲

常规

变量表达式的积与商

Writing Products and Quotients of Variable Expressions ::变量表达式的书写产品和引号

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Writing Products and Quotients of Variable Expressions
::变量表达式的书写产品和引号

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源