Course: 4.11简化包含整数乘法的变量表达式

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 简化包含整数乘法的变量表达式

Collapse Expand
简化包含整数乘法的变量表达式
Mr. Myers teaches geometry at Rosewood High School. The school year is about to start and he's thinking about how many tests he will have to grade over the course of the year. He isn't sure exactly how many tests he will be giving, but each of his classes has 25 students and he has 5 geometry classes. If $\begin{aligned} t \end{aligned}$ represents the number of tests he gives each student that year, how can Mr. Myers write and simplify an expression to represent the total number of tests he will have to grade over the course of the year?
::迈尔斯先生在罗斯伍德高中教授几何学。学年即将开始,他正在考虑在一年中要进行多少次考试。他不确定要进行多少次考试,但他不确定要进行多少次考试,但他每个班都有25名学生,他有5个几何班。如果不能代表他每年给每个学生的考试数量,那么迈尔斯先生怎么写和简化表达方式来表示他一年中必须进行的考试总数?

In this concept, you will learn how to simplify terms within variable expressions using integer multiplication .
::在此概念中,您将学习如何使用整数乘法简化变量表达式中的术语。

Simplifying Variable Expressions Involving Integer Multiplication
::简化包含整数乘法的变量表达式

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 .
::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隔的术语组成。

Here is an example:
::以下是一个例子:

$\begin{aligned} 2 x \cdot (- 4) + 3 x - 1 \end{aligned}$

In this variable expression there are 3 terms. The first term is $\begin{aligned} 2 x \cdot (- 4) \end{aligned}$ , the second term is $\begin{aligned} 3 x \end{aligned}$ , and the third term is -1.
::在这个可变表达式中,有三个术语。第一个术语是 2 x ( - 4 ) ,第二个术语是 3 x ,第三个术语是 - 1 。

Sometimes individual terms can be simplified if they contain more than one number.
::有时,个别术语如果包含一个以上的数字,可以简化。

For example, in the variable expression above, the term $\begin{aligned} 2 x \cdot (- 4) \end{aligned}$ can be simplified. Let's look at how you would simplify that term.
::例如,在上文变量表达式中,2 x (-4) 术语可以简化。让我们看看您如何简化该术语。

The first step is to rewrite $\begin{aligned} 2 x \cdot (- 4) \end{aligned}$ as $\begin{aligned} (- 4) \cdot 2 x \end{aligned}$ .
::第一步是将 2 x (-4 ) 改写为 (-4 ) 2 x 。

You can flip the order of the multiplication due to the commutative property of multiplication which states that the order in which factors are multiplied does not matter.
::您可以翻转乘法的顺序,因为乘法的通量属性表明乘法的顺序无关紧要。

The next step is to group the integers together.
::下一步是将整数组合在一起。

Rewrite $\begin{aligned} (- 4) \cdot 2 x \end{aligned}$ as $\begin{aligned} (- 4 \cdot 2) \cdot x \end{aligned}$ .
::重写 ( - 4) 2 x 值 ( - 4 ) 2 x 值 ( - 4 2 ) x 。

You can change the way the parts of the term are grouped due to the associative property of multiplication which states that you can group the factors being multiplied in any order.
::您可以因乘法的关联属性而改变该词各部分的组合方式,该属性表示您可以按任何顺序将乘以的系数组合在一起。

Then, you can multiply the integers. Remember that a negative times a positive equals a negative, so:
::然后,您可以乘以整数。记住负乘以正负等于负,所以:

$\begin{aligned} - 4 \cdot 2 = - 8 \end{aligned}$

$\begin{aligned} (- 4 \cdot 2) \cdot x = - 8 x \end{aligned}$

The answer is $\begin{aligned} 2 x \cdot (- 4) = - 8 x \end{aligned}$ .
::答案是 2 x ( - 4 ) = - 8 x 。

Keep in mind that whenever you are simplifying a term, you are combining all the integers in the term that are being multiplied by performing the multiplication. When you write your answer, the variables that were in the term at the beginning will appear at the end of the term.
::请注意,当您正在简化一个术语时,您正在将该术语中通过执行乘法而乘以乘以的所有整数合并在一起。当您写入您的答复时,该术语开头的变量将出现在该术语的末尾。

Let's look at another example.
::让我们再举一个例子。

Simplify "> $\begin{aligned} (- 5) (- 2 m) (n) \end{aligned}$ .
::简化(-5) (-2米) 。

This variable expression has only one term. It can be simplified by combining the integers within the term together through multiplication.
::此变量表达式只有一个术语。可以通过乘法将术语内的整数合并来简化它。

First, rewrite this expression by grouping all of the integers together and leaving the variables at the end.
::首先,重写此表达式,将全部整数组合在一起,并将变量留在结尾处。

%3D(%5Ctext%7B-%7D5%20%5Ccdot%20%5Ctext%7B-%7D2)mn"> $\begin{aligned} (- 5) (- 2 m) (n) = (- 5 \cdot - 2) m n \end{aligned}$
::- 5) (-2米) = (-5) - 2) n

Next, multiply the integers so that you have one integer in your expression instead of two. Remember that a negative times a negative equals a positive.
::下一位, 乘以整数, 这样表达式中就有一个整数, 而不是两个整数。记住负乘以负等于正数。

$\begin{aligned} - 5 \cdot - 2 = 10 \end{aligned}$

$\begin{aligned} (- 5 \cdot - 2) m n = 10 m n \end{aligned}$
::- 5 - 2 ) m n = 10 m n = 10 m n

The answer is %3D10mn"> $\begin{aligned} (- 5) (- 2 m) (n) = 10 m n \end{aligned}$ .
::答案是(-5) (-2米) = 10米。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Mr. Myers and his geometry classes.
::早些时候,有人给了你一个问题关于迈尔斯先生和他的几何等级。

He teaches 5 geometry classes and each class has 25 students. If he gives each student $\begin{aligned} t \end{aligned}$ tests during the school year, how many tests will he have to grade over the course of the year?
::他教5个几何课,每个班有25名学生。如果他在学年里给每个学生做 t 测试,那么在一年中他还要给多少次测试分级?

If $\begin{aligned} t \end{aligned}$ represents the number of tests that he gives each student, then the number of tests he will have to grade from one geometry class is $\begin{aligned} 25 t \end{aligned}$ .
::如果 t 表示他给每个学生的测试次数,那么他从一个几何等级中得分的测试次数为 25 吨。

He teaches 5 geometry classes, so the total number of tests he will have to grade is:
::他教5个几何等级, 所以他要分级的测试总数是:

$\begin{aligned} (25 t) \cdot 5 \end{aligned}$
:25吨) 5

You can simplify this expression by combining the integers together through multiplication.
::您可以通过乘法将整数合并来简化这个表达式。

First, rewrite the expression by grouping all of the integers together and leaving the variable at the end.
::首先,重写表达式,将全部整数组合在一起,然后将变量留在结尾处。

$\begin{aligned} (25 t) \cdot 5 = (25 \cdot 5) t \end{aligned}$
:25吨) 5 = (25吨) 5 = (25吨) 5 t

Next, multiply the integers so that you have one integer in your expression instead of two.
::下一个,乘以整数,使表达式中有一个整数,而不是两个整数。

$\begin{aligned} 25 \cdot 5 = 125 \end{aligned}$

$\begin{aligned} (25 \cdot 5) t = 125 t \end{aligned}$
:25 5 ) t = 125 t

The answer is $\begin{aligned} (25 t) \cdot 5 = 125 t \end{aligned}$ .
::答案是(25吨) 5 = 125吨。

Mr. Myers will have to grade $\begin{aligned} 125 t \end{aligned}$ tests over the course of the year.
::迈尔斯先生将不得不在一年中完成125吨的考试。

Example 2
::例2

Simplify $\begin{aligned} - 2 x (4 y) (6) \end{aligned}$ .
::简化 - 2 x (4 y) (6 ) 。

First, rewrite this expression by grouping all of the integers together and leaving the variables at the end.
::首先,重写此表达式,将全部整数组合在一起,并将变量留在结尾处。

$\begin{aligned} - 2 x (4 y) (6) = (- 2 \cdot 4 \cdot 6) x y \end{aligned}$
::- 2 x (4 y) (6 ) = (-2 + 4 + 6) x y

Next, multiply the integers so that you have one integer in your expression instead of three. You can do this in two steps. First multiply $\begin{aligned} - 2 \cdot 4 \end{aligned}$ and then multiply the result by 6.
::下一步,乘以整数,使您的表达式中有一个整数,而不是三个整数。您可以分两个步骤做到这一点。首先乘以 - 2 + 4,然后将结果乘以 6。

$\begin{aligned} - 2 \cdot 4 = - 8 \end{aligned}$

$\begin{aligned} - 8 \cdot 6 = - 48 \end{aligned}$

$\begin{aligned} (- 2 \cdot 4 \cdot 6) x y = - 48 x y \end{aligned}$
:- 2 + 4 + 6 ) x y = - 48 x y)

The answer is $\begin{aligned} - 2 x (4 y) (6) = - 48 x y \end{aligned}$ .
::回答是- 2 x (4 y) (6) = - 48 x y 。

Example 3
::例3

Simplify $\begin{aligned} 3 x (4 y) \end{aligned}$ .
::简化 3 x (4 y) 。

First, rewrite this expression by grouping all of the integers together and leaving the variables at the end.
::首先,重写此表达式,将全部整数组合在一起,并将变量留在结尾处。

$\begin{aligned} 3 x (4 y) = (3 \cdot 4) x y \end{aligned}$
::3 x (4 y) = (3 + 4) x y

Next, multiply the integers so that you have one integer in your expression instead of two.
::下一个,乘以整数,使表达式中有一个整数,而不是两个整数。

$\begin{aligned} 3 \cdot 4 = 12 \end{aligned}$

$\begin{aligned} (3 \cdot 4) x y = 12 x y \end{aligned}$
:3 + 4 ) x y = 12 x y y)

The answer is $\begin{aligned} 3 x (4 y) = 12 x y \end{aligned}$ .
::答案是 3 x (4 y) = 12 x y 。

Example 4
::例4

Simplify $\begin{aligned} - 6 a (- 4 b) \end{aligned}$ .
::简化 - 6 a (- 4 b) 。

First, rewrite this expression by grouping all of the integers together and leaving the variables at the end.
::首先,重写此表达式,将全部整数组合在一起,并将变量留在结尾处。

$\begin{aligned} - 6 a (- 4 b) = (- 6 \cdot - 4) a b \end{aligned}$
::- 6 a (- 4 b) = (- 6 - 4) a b

Next, multiply the integers so that you have one integer in your expression instead of two.
::下一个,乘以整数,使表达式中有一个整数,而不是两个整数。

$\begin{aligned} - 6 \cdot - 4 = 24 \end{aligned}$

$\begin{aligned} (- 6 \cdot - 4) a b = 24 a b \end{aligned}$
:- 6 - 4) a b = 24 a b

The answer is $\begin{aligned} - 6 a (- 4 b) = 24 a b \end{aligned}$ .
::答案是 - 6 a (- 4 b) = 24 a b 。

Example 5
::例5

Simplify $\begin{aligned} - 4 z (10) \end{aligned}$ .
::简化 - 4 z( 10 ) 。

First, rewrite this expression by grouping all of the integers together and leaving the variable at the end.
::首先,重写此表达式,将全部整数组合在一起,然后将变量留在结尾处。

$\begin{aligned} - 4 z (10) = (- 4 \cdot 10) z \end{aligned}$
::- 4z(10)=(4-10)z

Next, multiply the integers so that you have one integer in your expression instead of two.
::下一个,乘以整数,使表达式中有一个整数,而不是两个整数。

$\begin{aligned} - 4 \cdot 10 = - 40 \end{aligned}$

$\begin{aligned} (- 4 \cdot 10) z = - 40 z \end{aligned}$
:- 4 10 ) z = - 40 z

The answer is $\begin{aligned} - 4 z (10) = - 40 z \end{aligned}$ .
::答案是- 4z(10) =- 40z。

Review
::回顾

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

$\begin{aligned} (- 7 k) (- 6) \end{aligned}$
:-7 k) (-6))

$\begin{aligned} (- 8) (3 a) (b) \end{aligned}$
:-8) (3a) (b) (8) (3a) (b)

$\begin{aligned} - 6 a (b) (c) \end{aligned}$
::- 6 a (b) (c) - 6 a (b) (c)

$\begin{aligned} - 8 a (6 b) \end{aligned}$
::8 a (6 b) - 8 a (6 b)

$\begin{aligned} (12 y) (- 3 x) (- 1) \end{aligned}$
:12 y) (- 3 x) (- 1))

$\begin{aligned} - 8 x (4) \end{aligned}$
::8 x (4 ) - 8 x (4 )

$\begin{aligned} - a (5) (- 4 b) \end{aligned}$
::- a (5) (- 4 b)

$\begin{aligned} - 2 a b (12 c) \end{aligned}$
::- 2 a b (12 c)

$\begin{aligned} - 12 a b (12 c) \end{aligned}$
::- 12个a b (12个c)

$\begin{aligned} 8 x (12 z) \end{aligned}$
::8 x( 12 z )

$\begin{aligned} - 2 a (- 14 c) \end{aligned}$
::- 2 a (- 14 c)

$\begin{aligned} - 12 a b (11 c) \end{aligned}$
::- 12ab(11c)

$\begin{aligned} - 22 a b (- 2 c) \end{aligned}$
::- 22 a b (-2 c)

$\begin{aligned} 18 a b (12) \end{aligned}$
::18ab(12) 18ab(12)

$\begin{aligned} - 21 a (- 3 b) \end{aligned}$
::- 21 a (- 3 b)

Review (Answers)
::回顾(答复)

Section outline

General

简化包含整数乘法的变量表达式

Simplifying Variable Expressions Involving Integer Multiplication ::简化包含整数乘法的变量表达式

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 Multiplication
::简化包含整数乘法的变量表达式

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)