Course: 4.16表达式和分配律

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表达式和分配律
Karen's biology class is going on a trip to the beach. Karen is in charge of keeping track of who has paid for the trip and who still needs to pay. There are 50 students going on the trip. Each student must pay $5 for transportation, $4.50 for lunch, and $3 for a t-shirt. How could Karen write and simplify a numerical expression using the to determine how much money should be collected total?
::Karen的生物学课正在去海滩旅行。 Karen负责追踪谁支付了旅费,谁还需要支付费用。有50名学生正在旅行中。每个学生必须支付5美元的交通费,4.50美元的午餐费,3美元的T恤费。 Karen如何用数字表达来写和简化数字表达方式,以决定应该收取多少钱总额?

In this concept, you will learn how to use the distributive property to write and evaluate equivalent numerical and variable expressions.
::在此概念中,您将学会如何使用分配属性来写写和评估等值的数字表达式和变量表达式。

Using the Distributive Property with Expressions
::使用含有表达式的分配属性

The distributive property states that if a factor is multiplied by the sum of two numbers, you can multiply each of the two numbers by that factor and then add them to produce the same result. In symbols, the distributive property says that for numbers $\begin{aligned} a \end{aligned}$ , $\begin{aligned} b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ :
::分配财产表示,如果一个系数乘以两个数字的总和,您可以将两个数字中的每一个乘以该系数,然后添加它们来产生同样的结果。在符号中,分配财产表示,对于数字a、b和c,分配财产表示:

$\begin{aligned} a (b + c) = a b + a c \end{aligned}$
::a (b) + c) = a b + a c

Anytime you see a number outside of a set of " data-term="Parentheses" role="term" tabindex="0"> parentheses within an expression, you can use the distributive property to help you to simplify by writing an equivalent expression without parentheses.
::当您在表达式中看到一组括号外的多个数字时,您可以使用分配属性来帮助您简化,在括号中没有括号的情况下写出等值表达式。

Let's look at an example.
::让我们举个例子。

$\begin{aligned} 5 (2 + 3) \end{aligned}$

You want to simplify by writing an equivalent expression without parentheses.
::您想要通过写一个没有括号的等同表达式来简化。

First, multiply the 5 by each of the terms inside the parentheses.
::首先,将括号内每个用语乘以5。

$\begin{aligned} 5 (2 + 3) = 5 (2) + 5 (3) \end{aligned}$

Next, simplify each part of the expression and then add.
::接下来,简化表达式的每个部分,然后添加。

$\begin{aligned} 5 (2) + 5 (3) = 10 + 15 = 25 \end{aligned}$

The answer is $\begin{aligned} 5 (2 + 3) = 25 \end{aligned}$ .
::答案是5(2+3)=25。

The distributive property also works if there is subtraction inside the parentheses. This is because subtracting a positive is just like adding a negative.
::如果括号内有减法,分配财产也会起作用。这是因为减去正数就等于加上负数。

Here is an example.
::举一个例子。

$\begin{aligned} 3 (3 - 2) \end{aligned}$

You want to simplify by writing an equivalent expression without parentheses.
::您想要通过写一个没有括号的等同表达式来简化。

First, multiply the 3 by each of the terms inside the parentheses.
::首先,将括号内每个用语乘以3。

$\begin{aligned} 3 (3 - 2) = 3 (3) - 3 (2) \end{aligned}$

Next, simplify each part of the expression and then subtract.
::下一步,简化表达式的每个部分,然后减去。

$\begin{aligned} 3 (3) - 3 (2) = 9 - 6 = 3 \end{aligned}$

The answer is $\begin{aligned} 3 (3 - 2) = 3 \end{aligned}$ .
::答案是3 (3 - 2) = 3 。

Notice this is the same answer that you would have gotten had you first subtracted the numbers inside the parentheses and then multiplied:
::请注意,这是同样的答案,如果你先减去括号内的数字,然后乘以:

$\begin{aligned} 3 (3 - 2) = 3 (1) = 3 \end{aligned}$

The distributive property isn't as necessary in situations where all parts of your expressions are numbers like in the previous two examples. However, it becomes very useful with variable expressions that contain variables. The distributive property helps you to get rid of the parentheses in variable expressions.
::当您的表达式的所有部分都是像前两个示例中的数字时,分配属性就没有必要了。然而,它对于包含变量的变量表达式非常有用。分配属性有助于您删除变量表达式中的括号。

Here is an example.
::举一个例子。

$\begin{aligned} 4 (x + 3) \end{aligned}$
::4(xx+3))

First, write an equivalent expression without parentheses. Multiply the 4 by each term inside the parentheses.
::首先,在括号中写出同等的表达式,无括号。将括号内四个词乘以每个词。

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

Next, simplify each part of the expression.
::接下来,简化表达式的每个部分。

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

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

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Karen and her biology class trip to the beach.
::早些时候,你得到一个问题关于凯伦和她的生物学课出海海滩。

Each of the 50 students going on the trip must pay $5 for transportation, $4.50 for lunch, and $3 for a t-shirt. Karen wants to know how much money will be collected in total.
::参加这次旅行的50名学生每人必须支付5美元的交通费、4.50美元的午餐费和3美元的T恤费。 Karen想知道总共将收取多少钱。

Each of the 50 students going on the trip has to pay $\begin{aligned} 5 + 4.50 + 3 \end{aligned}$ for transportation, lunch, and a t-shirt. That means the total amount of money collected will be
::旅行的50名学生中,每人必须支付5+4.50+3的交通费、午餐费和一件T恤衫。

$\begin{aligned} 50 (5 + 4.50 + 3) \end{aligned}$

One way to complete this calculation is by using the distributive property.
::完成这一计算的方法之一是使用分配财产。

First, write an equivalent expression without parentheses. Multiply the 5 by each of the terms inside the parentheses.
::首先,请在无括号的情况下写出等值表达式。将括号内的5个词乘以括号内的每个词。

$\begin{aligned} 50 (5 + 4.50 + 3) = 50 (5) + 50 (4.50) + 50 (3) \end{aligned}$

Next, simplify each part of the expression and then add.
::接下来,简化表达式的每个部分,然后添加。

$\begin{aligned} 50 (5) + 50 (4.50) + 50 (3) = 250 + 225 + 150 = 625 \end{aligned}$

The answer is 625.
::答案是625

In total, $625 will be collected for the trip to the beach.
::沙滩旅行将共收取625美元。

Example 2
::例2

Liam has a rectangular backyard that is 20 yards long and 18 yards wide. He wants to use a part of his yard that is 20 yards by 8 yards for a vegetable garden. If he does this, what will be the area of the section of the yard that will not be used as a garden?
::Liam有一个长20码、宽18码的长方形后院。他想用院子的20码乘8码的院子作为菜园。如果他这样做,那么院子那部分不会被用作花园的面积会是多少?

First, remember that to find the area of a rectangle you need to multiply length times width.
::首先, 请记住, 要找到矩形区域, 您需要将长度乘以宽度。

Next, figure out the length and width of the part of the backyard not used as a garden.
::接下来,找出后院的长度和宽度不用作花园的部分。

The length of the part of the backyard not used as a garden is 20 yards.
::不用作花园的后院部分长度为20码。

The width of the part of the backyard not used as a garden is (18 yards - 8 yards).
::未用作花园的后院的宽度为18码至8码。

Now, write out the product of length times width.
::现在,写出长度乘以宽度的产物。

$\begin{aligned} Area = 20 (18 - 8) \end{aligned}$
::面积=20(18-8)

One way to complete this multiplication is by using the distributive property. Multiply 20 by each term inside the parentheses.
::完成这一乘法的一个方法是使用分配属性。括号内每个词乘以20。

$\begin{aligned} Area = 20 (18) - 20 (8) \end{aligned}$
::面积=20(18)-20(8)

Now simplify each part of the expression and then subtract.
::现在简化表达式的每个部分,然后减去。

$\begin{aligned} Area = 360 - 160 = 200 \end{aligned}$
::面积=360 - 160 = 200

The answer is that the area of the backyard not used as a garden is 200 square yards.
::答案是,后院内没有作为花园使用的面积是200平方码。

Example 3
::例3

Simplify using the distributive property.
::使用分配性财产简化。

$\begin{aligned} 6 (5 + 2) \end{aligned}$

First, write an equivalent expression without parentheses. Multiply the 6 by each of the terms inside the parentheses.
::首先,请在无括号的情况下写出等值表达式。将括号内的6个词乘以括号内的每个词。

$\begin{aligned} 6 (5 + 2) = 6 (5) + 6 (2) \end{aligned}$

Next, simplify each part of the expression and then add.
::接下来,简化表达式的每个部分,然后添加。

$\begin{aligned} 6 (5) + 6 (2) = 30 + 12 = 42 \end{aligned}$

The answer is $\begin{aligned} 6 (5 + 2) = 42 \end{aligned}$ .
::答案是6(5+2)=42。

Example 4
::例4

Simplify using the distributive property.
::使用分配性财产简化。

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

First, write an equivalent expression without parentheses. Multiply the 3 by each term inside the parentheses.
::首先,在括号中写出同等的表达式,无括号。将括号内的三个词乘以每个词。

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

Next, simplify each part of the expression.
::接下来,简化表达式的每个部分。

$\begin{aligned} 3 (x) - 3 (5) = 3 x - 15 \end{aligned}$
::3 (xx) - 3 (5) = 3 x - 15

The answer is $\begin{aligned} 3 (x - 5) = 3 x - 15 \end{aligned}$ .
::答案是3(x-5)=3x-15。

Example 5
::例5

Simplify using the distributive property.
::使用分配性财产简化。

$\begin{aligned} 8 (9 + y) \end{aligned}$
::8(9+y)

First, write an equivalent expression without parentheses. Multiply the 8 by each term inside the parentheses.
::首先,在括号中写出同等的表达式,无括号。将括号内每个词的8乘以8。

"> $\begin{aligned} 8 (9 + y) = 8 (9) + 8 (y) \end{aligned}$
::8(9+y)=8(9)+8

Next, simplify each part of the expression.
::接下来,简化表达式的每个部分。

%3D72%2B8y"> $\begin{aligned} 8 (9) + 8 (y) = 72 + 8 y \end{aligned}$
::8(9)+8=72+8y

The answer is $\begin{aligned} 8 (9 + y) = 72 + 8 y \end{aligned}$ .
::答案是8(9+y)=72+8y。

Review
::回顾

Use the distributive property to write an equivalent expression for each numerical expression.
::使用分配属性为每个数字表达式写入等值表达式。

$\begin{aligned} 6 (3 + 4) \end{aligned}$

$\begin{aligned} 5 (4 + 1) \end{aligned}$

$\begin{aligned} 12 (3 + 5) \end{aligned}$

$\begin{aligned} 6 (7 + 8) \end{aligned}$

$\begin{aligned} 2 (4 + 5) \end{aligned}$

$\begin{aligned} 3 (5 - 2) \end{aligned}$

$\begin{aligned} 6 (7 - 3) \end{aligned}$

$\begin{aligned} 5 (4 - 2) \end{aligned}$

$\begin{aligned} 7 (5 - 1) \end{aligned}$

$\begin{aligned} 6 (9 - 3) \end{aligned}$

Use the distributive property to write an equivalent expression for each variable expression.
::使用分配属性为每个变量表达式写入等效表达式。

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

$\begin{aligned} 6 (y - 2) \end{aligned}$
::6 y y - 2 )

$\begin{aligned} 5 (x + 9) \end{aligned}$
::5(x+9) 5(x+9)

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

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

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

Section outline

General

表达式和分配律

Using the Distributive Property with Expressions ::使用含有表达式的分配属性

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

Using the Distributive Property with Expressions
::使用含有表达式的分配属性

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