Course: 4.13简化包含整数除法的变量表达式

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简化包含整数除法的变量表达式
Jake and his 3 friends are working together to sell bird houses that they've made. They are selling the bird houses for $20 each and plan to divide up the money they make equally. If $\begin{aligned} b \end{aligned}$ represents the number of bird houses they sell, how could Jake write and simplify a variable expression that represents how much money he will make in terms of $\begin{aligned} b \end{aligned}$ ?
::Jake和他的三个朋友正在共同努力出售他们建造的鸟屋。他们每家鸟屋出售20美元,并计划平均分配它们赚的钱。如果b表示它们卖的鸟屋数量,Jake怎么能写和简化一个代表他能赚多少钱的变数表达方式?

In this concept, you will learn how to simplify terms within variable expressions using integer division .
::在此概念中,您将学会如何使用整数分隔来简化变量表达式中的术语。

Simplifying Variable Expressions Involving Integer Division
::涉及整数分区的简化变量表达式

Recall that a variable expression is a math phrase that has numbers, variables, and operations in it. Variable expressions are made up of terms that are separated by addition or subtraction .
::回顾变量表达式是一个数学短语,其中含有数字、变量和操作。变量表达式由增加或减法分开的术语组成。

Here is an example:
::以下是一个例子:

$\begin{aligned} \frac{24 x}{3} + 18 x y \end{aligned}$
::24 x 3 + 18 x y

In this variable expression there are two terms. The first term is $\begin{aligned} \frac{24 x}{3} \end{aligned}$ and the second term is $\begin{aligned} 18 x y \end{aligned}$ .
::在此变量表达式中有两个术语。第一个术语为 24 x 3, 第二个术语为 18 x y 。

Sometimes individual terms can be simplified if they contain more than one number or the same variable more than once.
::有时,个别术语如果不止一次包含一个数字或同一个变量,则可以简化。

For example, in the variable expression above, the term $\begin{aligned} \frac{24 x}{3} \end{aligned}$ can be simplified. Let's look at how you would simplify it.
::例如,在以上变量表达式中,24 x 3 术语可以简化。让我们看看如何简化它。

Remember that a fraction bar is the same as division. So $\begin{aligned} \frac{24 x}{3} \end{aligned}$ is the same as $\begin{aligned} 24 x \div 3 \end{aligned}$ . You will see expressions written in both ways. If the expression is not already in fraction form, it helps to rewrite it in fraction form.
::记住分数栏与分数栏相同。所以 24 x 3 和 24 x {{{{{{} 3 相同。您将会看到以两种方式书写的表达式。如果表达式尚未以分数形式出现, 则有助于以分数形式重写它。

$\begin{aligned} \frac{24 x}{3} \end{aligned}$
::24 x 3 24 x 3

Your next step is to separate out the integers and variable in the numerator and the denominator.
::您的下一步是分离分子和分母中的整数和变量。

$\begin{aligned} \frac{24 x}{3} = \frac{24 \cdot x}{3} \end{aligned}$
::24 x 3 = 24 x 3

Now, divide the integers within the term.
::现在,在术语内将整数除以。

$\begin{aligned} \frac{24}{3} = 8 \end{aligned}$

There is only one $\begin{aligned} x \end{aligned}$ in the expression, so it will not change.
::表达式中只有一个 x, 所以它不会改变。

The answer is $\begin{aligned} \frac{24 x}{3} = 8 x \end{aligned}$ .
::答案是 24 x 3 = 8 x 。

Let's look at another example where there is the same variable more than once.
::让我们再看看另一个例子,其中同一个变量不止一次。

Simplify $\begin{aligned} - 24 y \div 2 y \end{aligned}$ .
::简化 - 24 y = 2 y 。

First, rewrite using a fraction bar.
::首先,使用分数条重写。

$\begin{aligned} - 24 y \div 2 y = \frac{- 24 y}{2 y} \end{aligned}$
::- 24 y = 2 y = - 24 y 2 y

Next, separate out the integers and variables in the numerator and the denominator.
::下一步,将分子和分母中的整数和变量分开。

$\begin{aligned} \frac{- 24 y}{2 y} = \frac{- 24 \cdot y}{2 \cdot y} \end{aligned}$
::- 24 y y = 24 y 2 y y

Now, focus on the integers. Divide $\begin{aligned} \frac{- 24}{2} \end{aligned}$ . You know that $\begin{aligned} \frac{24}{2} = 12 \end{aligned}$ and a negative divided by a positive equals a negative.
::现在,关注整数。除以 - 24 2。你知道24 2 = 12, 负除以正数等于负。

$\begin{aligned} \frac{- 24}{2} = - 12 \end{aligned}$

Next, look at the variables. There is a $\begin{aligned} y \end{aligned}$ in both the numerator and the denominator. Any number divided by itself is equal to 1, so $\begin{aligned} \frac{y}{y} \end{aligned}$ is equal to 1. You might say that the $\begin{aligned} y \end{aligned}$ 's “cancel out”.
::接下来,请看变量。在分子和分母中都有一个y。任何数字本身除以等于 1, y y 等于 1。您可以说 y ' s " cancel out " 。

$\begin{aligned} \frac{- 24 \cdot y}{2 \cdot y} = - 12 \cdot 1 = - 12 \end{aligned}$
::- 24 y 2 y y = - 12 1 = 12

The answer is $\begin{aligned} - 24 y \div 2 y = - 12 \end{aligned}$ .
::答案是 24 y y y 2 y y = 12 。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Jake and his bird houses.
::早些时候,你得到一个问题关于杰克和他的鸟屋。

Jake and his three friends will sell $\begin{aligned} b \end{aligned}$ bird houses for $20 each and Jake wonders how much money he will make if he and his friends divide the money equally.
::Jake和他的三个朋友会卖B鸟屋每家20美元,Jake想知道如果他和他的朋友平分钱,他能赚多少钱。

If Jake and his friends sell $\begin{aligned} b \end{aligned}$ bird houses at $20 each, they will make 20b dollars total. They will divide up that money four ways amongst the four friends. Each person will get:
::如果Jake和他的朋友每人卖B鸟屋20美元,他们总共就能赚20美元。他们将把这笔钱分成四大类。

$\begin{aligned} \frac{20 b}{4} \end{aligned}$
::20 b 4

You can simplify this expression by dividing the integers. First, separate the integers from the variable:
::您可以通过分隔整数来简化此表达式。首先,将整数从变量中分离出来:

$\begin{aligned} \frac{20 \cdot b}{4} \end{aligned}$
::20 b 4

Now, focus on the integers. Divide $\begin{aligned} \frac{20}{4} \end{aligned}$ .
::现在,集中关注整数。除以 20 4 。

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

Next, look at the variable. There is only one $\begin{aligned} b \end{aligned}$ , so it will not change.
::接下来,看看变量。只有一个 b,所以它不会改变。

$\begin{aligned} \frac{20 \cdot b}{4} = 5 b \end{aligned}$
::20 b 4 = 5 b

The answer is $\begin{aligned} \frac{20 b}{4} = 5 b \end{aligned}$ .
::答案是20 b 4 = 5 b 。

Jake will make $\begin{aligned} 5 b \end{aligned}$ dollars from selling the bird houses.
::杰克会从卖鸟屋赚5美元

Example 2
::例2

Simplify $\begin{aligned} - 18 a b \div 9 b \end{aligned}$ .
::简化 - 18 a b = 9 b 。

First, rewrite using a fraction bar.
::首先,使用分数条重写。

$\begin{aligned} - 18 a b \div 9 b = \frac{- 18 a b}{9 b} \end{aligned}$
::- 18个a b 9 b = - 18个a b 9 b

Next, separate out the integers and variables in the numerator and the denominator.
::下一步,将分子和分母中的整数和变量分开。

$\begin{aligned} \frac{- 18 a b}{9 b} = \frac{- 18 \cdot a \cdot b}{9 \cdot b} \end{aligned}$
::- 18个a b 9 b = 18 a b 9 b

Now, focus on the integers. Divide $\begin{aligned} \frac{- 18}{9} \end{aligned}$ . You know that $\begin{aligned} \frac{18}{9} = 2 \end{aligned}$ and a negative divided by a positive equals a negative.
::现在,关注整数。除以 - 18 9 。你知道18 9 = 2 和负除以正数等于负。

$\begin{aligned} \frac{- 18}{9} = - 2 \end{aligned}$

Next, look at the variables. There is a $\begin{aligned} b \end{aligned}$ in both the numerator and the denominator. $\begin{aligned} \frac{b}{b} \end{aligned}$ is equal to 1. There is only one $\begin{aligned} a \end{aligned}$ , so it will not change.
::接下来,请看变量。分子和分母中都有一个 b 。b b 等于 1 。只有一个 a , 所以它不会改变。

$\begin{aligned} \frac{- 18 \cdot a \cdot b}{9 \cdot b} = - 2 \cdot a \cdot 1 = - 2 a \end{aligned}$
::- 18 a b b 9 b = 2 a = 1 = 2 a = 2 a = 2 a

The answer is $\begin{aligned} - 18 a b \div 9 b = - 2 a \end{aligned}$ .
::答案是 - 18 a b = 9 b = - 2 a 。

Example 3
::例3

Simplify $\begin{aligned} - 14 a \div - 7 \end{aligned}$ .
::简化 - 14 a * _ 7 。

First, rewrite using a fraction bar.
::首先,使用分数条重写。

$\begin{aligned} - 14 a \div - 7 = \frac{- 14 a}{- 7} \end{aligned}$
::- 14 a = 7 = - 14 a = 7 = 14 a = 7

Next, separate out the integers and variable in the numerator and the denominator.
::下一步,将分子和分母中的整数和变量分开。

$\begin{aligned} \frac{- 14 a}{- 7} = \frac{- 14 \cdot a}{- 7} \end{aligned}$
::- 14 a - 7 = - 14 a - 7

Now, focus on the integers. Divide $\begin{aligned} \frac{- 14}{- 7} \end{aligned}$ . You know that $\begin{aligned} \frac{14}{7} = 2 \end{aligned}$ and a negative divided by a negative equals a positive.
::现在,关注整数。除以 - 14 - 7。你知道14 - 7= 2,负除以负等于正。

$\begin{aligned} \frac{- 14}{- 7} = 2 \end{aligned}$

Next, look at the variable. There is only one $\begin{aligned} a \end{aligned}$ , so it will not change.
::接下来,看看变量。只有一个一个,所以它不会改变。

$\begin{aligned} \frac{- 14 \cdot a}{- 7} = 2 a \end{aligned}$
::- 14 a - 7 = 2 a

The answer is $\begin{aligned} - 14 a \div - 7 = 2 a \end{aligned}$ .
::答案是 - 14 a * _ - 7 = 2 a 。

Example 4
::例4

Simplify $\begin{aligned} 28 a b \div - 7 b \end{aligned}$ .
::简化 28 a b b b b = 7 b 。

First, rewrite using a fraction bar.
::首先,使用分数条重写。

$\begin{aligned} 28 a b \div - 7 b = \frac{28 a b}{- 7 b} \end{aligned}$
::28 a b - 7 b = 28 a b -7 b

Next, separate out the integers and variables in the numerator and the denominator.
::下一步,将分子和分母中的整数和变量分开。

$\begin{aligned} \frac{28 a b}{- 7 b} = \frac{28 \cdot a \cdot b}{- 7 \cdot b} \end{aligned}$
::28 a b - 7 b = 28 a b - 7 b

Now, focus on the integers. Divide $\begin{aligned} \frac{28}{- 7} \end{aligned}$ . You know that $\begin{aligned} \frac{28}{7} = 4 \end{aligned}$ and a positive divided by a negative equals a negative.
::现在,集中关注整数。除以 28 - 7。你知道, 28 7 = 4, 正除以负等于负。

$\begin{aligned} \frac{28}{- 7} = - 4 \end{aligned}$

Next, look at the variables. There is a $\begin{aligned} b \end{aligned}$ in both the numerator and the denominator. $\begin{aligned} \frac{b}{b} \end{aligned}$ is equal to 1. There is only one $\begin{aligned} a \end{aligned}$ , so it will not change.
::接下来,请看变量。分子和分母中都有一个 b 。b b 等于 1 。只有一个 a , 所以它不会改变。

$\begin{aligned} \frac{28 \cdot a \cdot b}{- 7 \cdot b} = - 4 \cdot a \cdot 1 = - 4 a \end{aligned}$
::28 ______________________________ b - 7 _____ b = - 4 _____ a 1 = 4 a = 4 a = 4 a = = 4 = = = = 4 = = = = = = = = = = = = = = = = = 4 = = = = = = = = = = = = = = = = = = = = = = = = = - = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

The answer is $\begin{aligned} 28 a b \div - 7 b = - 4 a \end{aligned}$ .
::答案是28 a b = 7 b = 4 a 。

Example 5
::例5

Simplify $\begin{aligned} - 6 x \div - 2 y \end{aligned}$ .
::简化 - 6 x - 2 y 。

First, rewrite using a fraction bar.
::首先,使用分数条重写。

$\begin{aligned} - 6 x \div - 2 y = \frac{- 6 x}{- 2 y} \end{aligned}$
::- 6x______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Next, separate out the integers and variables in the numerator and the denominator.
::下一步,将分子和分母中的整数和变量分开。

$\begin{aligned} \frac{- 6 x}{- 2 y} = \frac{- 6 \cdot x}{- 2 \cdot y} \end{aligned}$
::- 6 x - 2 y = - 6 + x - 2 y = - 6 + x - 2 y

Now, focus on the integers. Divide $\begin{aligned} \frac{- 6}{- 2} \end{aligned}$ . You know that $\begin{aligned} \frac{6}{2} = 3 \end{aligned}$ and a negative divided by a negative equals a positive.
::现在,关注整数。除以 - 6 - 2。你知道, 6 2 = 3, 负除以负等于正。

$\begin{aligned} \frac{- 6}{- 2} = 3 \end{aligned}$

Next, look at the variables. There is only one $\begin{aligned} x \end{aligned}$ , so it will not change. There is only one $\begin{aligned} y \end{aligned}$ , so it will not change.
::接下来,请看变量。只有一个 x,所以它不会改变。只有一个 y,所以它不会改变。

$\begin{aligned} \frac{- 6 \cdot x}{- 2 \cdot y} = \frac{3 x}{y} \end{aligned}$
::- 6 × x - 2 × y = 3 x y

Notice that the $\begin{aligned} y \end{aligned}$ must stay in the denominator of the fraction!
::注意你必须留在分数的分母中!

The answer is $\begin{aligned} - 6 x \div - 2 y = \frac{3 x}{y} \end{aligned}$ .
::答案是 - 6 x - 2 y = 3 x y 。

Review
::回顾

Simplify each variable expression.
::简化每个变量表达式。

$\begin{aligned} 36 t \div (- 9) \end{aligned}$
::36 t (- 9)

$\begin{aligned} - 56 n \div (- 7) \end{aligned}$
::- 56n ( - 7)

$\begin{aligned} - 22 n \div - 11 n \end{aligned}$
::- 22 n _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} - 28 n \div 7 \end{aligned}$
::- 28 n 7

$\begin{aligned} 18 x y \div 2 x \end{aligned}$
::18xy y 2x

$\begin{aligned} 72 t \div (- 9 t) \end{aligned}$
::72 t (- 9 t )

$\begin{aligned} 48 x y \div (- 8 y) \end{aligned}$
::48 x y ( - 8 y)

$\begin{aligned} 54 x y \div (- 9 x y) \end{aligned}$
::54xy (- 9xy)

$\begin{aligned} 16 a \div (4 a) \end{aligned}$
::16 a (4 a )

$\begin{aligned} - 16 a b \div (- 4 b) \end{aligned}$
::- 16 a b ( - 4 b)

$\begin{aligned} - 99 x y \div (- 9 x) \end{aligned}$
::- 99xy (- 9x)

$\begin{aligned} 121 a \div (11 b) \end{aligned}$
::121 a (11 b)

$\begin{aligned} - 144 x y \div (- 12) \end{aligned}$
::- 144xy ( - 12)

$\begin{aligned} - 169 y \div (- 13 x) \end{aligned}$
::- 169 y ( - 13 x )

$\begin{aligned} - 225 x y \div (5 z) \end{aligned}$
::- 225 x y (5 z )

Review (Answers)
::回顾(答复)

Section outline

General

简化包含整数除法的变量表达式

Simplifying Variable Expressions Involving Integer Division ::涉及整数分区的简化变量表达式

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)

Simplifying Variable Expressions Involving Integer Division
::涉及整数分区的简化变量表达式

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)