Course: 1.3 代数属性 | The Way To Learn

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代数属性

Abigail buys 2 shirts at $12 each . She also buys a pair of jeans that were originally $50 but are discounted by 15%. If her total purchase is over $60, she will get an additional $10 off. How much was her bill?
::艾比盖尔每人买两件衬衫,每件12美元。她还买一双牛仔裤,原来50美元,但折扣15 % 。如果她的购买总额超过60美元,她将多拿10美元。她的帐单多少钱?

Algebra Properties
::代数属性

The properties of algebra enable us to solve mathematical equations. Notice that these properties hold for addition and multiplication .
::代数的特性使我们能够解析数学方程。注意这些属性包含添加和乘法。

Property	Example
Commutative	$\begin{aligned} a + b = b + a, a b = b a \end{aligned}$
Associative	$\begin{aligned} a + (b + c) = (a + b) + c, a (b c) = (a b) c \end{aligned}$
Identity	$\begin{aligned} a + 0 = a, a \cdot 1 = a \end{aligned}$
Inverse	$\begin{aligned} a + (- a) = 0, a \cdot \frac{1}{a} = 1 \end{aligned}$
Distributive	$\begin{aligned} a (b + c) = a b + a c \end{aligned}$

From the Identity Property , we can say that 0 is the additive identity and 1 is the multiplicative identity . Similarly, from the Inverse Property, $\begin{aligned} - a \end{aligned}$ is the additive inverse of $\begin{aligned} a \end{aligned}$ and $\begin{aligned} \frac{1}{a} \end{aligned}$ is the multiplicative inverse of $\begin{aligned} a \end{aligned}$ because they each equal the associated identity.
::从身份财产中,我们可以说0是附加身份,1是倍增效应身份。同样,从反向财产中,a是a的复数反差,a是a的复数反差,a是a的复数反差,因为两者都等于相关身份。

Identify the property used in the equation : $\begin{aligned} 2 (4 x - 3) = 8 x - 6 \end{aligned}$
::方程中使用的属性标识: 2( 2( 4x- 3) = 8x- 6

This is an example of the .
::这是一个例子。

Identify the property used in the equation: $\begin{aligned} 5 \cdot \frac{1}{5} = 1 \end{aligned}$
::方程中使用的财产: 515=1

This is an example of the Inverse Property.
::这是反向财产的例子。

Identify the property used in the equation: $\begin{aligned} 6 \cdot (7 \cdot 8) = (6 \cdot 7) \cdot 8 \end{aligned}$
::方程中所用财产的识别: 6(78)=(67)8

This is an example of the Associative Property .
::这是共同财产的一个例子。

Going along with the properties of algebra is the Order of Operations. The Order of Operations is a set of guidelines that allow mathematicians to perform problems in the exact same way. The order is as follows:
::与代数的特性相配合的是《行动命令》,《行动命令》是一套准则,允许数学家以同样的方式处理问题。

Parenthesis: Do anything in parenthesis first.
::括号:先做括号中的任何事项。

Exponents: Next, all powers (exponents) need to be evaluated.
::指数:接下来,所有权力(功率)都需要评估。

Multiplication/ Division : Multiplication and division are done at the same time, from left to right.
::乘法/司:乘法和除法同时进行,从左向右。

Addition/ Subtraction : Addition and subtraction are also done together, from left to right.
::加/减:加和减也一起进行,从左到右。

Simplify using PEMDAS $\begin{aligned} 2^{2} + 6 \cdot 3 - (5 - 1) \end{aligned}$ .
::使用 PEMDAS 22+63-(5-1)进行简化。

First: Parenthesis $\begin{aligned} \to 2^{2} + 6 \cdot 3 - 4 \end{aligned}$
::第一:括号 22+63-4

Next: Exponents $\begin{aligned} \to 4 + 6 \cdot 3 - 4 \end{aligned}$
::下一位:4+6+63-4的指数

Then: Multiplication $\begin{aligned} \to 4 + 18 - 4 \end{aligned}$
::然后:乘数%4+18-4

Finally: Add/Subtract $\begin{aligned} \to 18 \end{aligned}$
::最后:加/减_____________________________________________________________________________________________________

Simplify using PEMDAS $\begin{aligned} \frac{9 - 4 \div 2 + 13}{2^{2} \cdot 3 - 7} \end{aligned}$ .
::使用 PEMDAS 9-42+1322_3-7 进行简化。

Think of everything in the numerator as being in its own set of parenthesis, and everything in the denominator as being in another set. The problem can be rewritten as $\begin{aligned} (9 - 4 \div 2 + 13) \div (2^{2} \cdot 3 - 7) \end{aligned}$ . When there are multiple operations in a set of parenthesis, use the Order of Operations within each set.
::将分子中的所有事物都想象成属于自己的括号, 分母中的所有事物都想象成属于另一组。问题可以重写为 (9 - 42+13)\\ (22_ 3-7) 。当在一组括号中有多个操作时, 请在每组内使用操作顺序。

\begin{aligned} (9 - 4 \div 2 + 13) \div (2^{2} \cdot 3 - 7) \\ (9 - 2 + 13) \div (4 \cdot 3 - 7) \\ (7 + 13) \div (12 - 7) \\ 20 \div 5 \\ 4 \end{aligned}

Parenthesis can also be written within another set of parenthesis. This is called embedding parenthesis. When embedded parenthesis are in a problem, you may see brackets , [ ], in addition to parenthesis.
::括号也可以写在另一组括号中。这被称为嵌入括号。当嵌入括号有问题时, 您除了括号外, 还可以看到括号 [ ] 。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the total cost of Abigail's bill.
::早些时候,你被要求找出Abigail账单的总成本。

Without knowing the Order of Operations, Abigail could get very confused and not have the right amount of money to pay for her items. So, the two shirts are $24 $\begin{aligned} (2 \cdot $ 12) \end{aligned}$ and only the jeans are discounted. They are $\begin{aligned} 50 - 0.15 \cdot 50 = 42.50 \end{aligned}$ . Adding these two items together, we get that her total bill is $24 + $42.50 = $66.50. So, because her purchase is over $60, she gets an additional $10 off, making her final bill $56.50.
::Abigail在不知道行动令的情况下,可能会变得非常混乱,没有适当的钱来支付她的物品。所以,两件衬衫是24美元(2-12美元),只有牛仔裤被打折扣。它们是50-0.15-50=42.50。加上这两件,我们得知她的总帐单是24美元+42.50=66.50美元。因此,由于她购买了60多美元,她又少了10美元,最后的帐单是56.50美元。

Example 2
::例2

What property is being used?
::使用哪些财产?

$\begin{aligned} 5 (c - 9) = 5 c - 45 \end{aligned}$
::5(c-9)=5c-45

5 is being distributed to each term inside the parenthesis, therefore the Distributive Property is being used.
::括号内有5个术语,正分配给每个术语,因此,正在使用分配财产。

Example 3
::例3

What property is being used?
::使用哪些财产?

$\begin{aligned} 6 \cdot 7 = 7 \cdot 6 \end{aligned}$

Here, order does not matter when multiplying 6 and 7. This is an example of the Commutative Property.
::这里,在乘以6和7时,顺序无关紧要。这是公用财产的例子。

Example 4
::例4

Use the Order of Operations to simplify $\begin{aligned} 8 + [4^{2} - 6 \div (5 + 1)] \end{aligned}$ .
::使用操作顺序简化 8+[42-6(5+1)]。

This is an example of embedded parenthesis, as discussed above. Start by simplifying the parenthesis that are inside the brackets. Then, simplify what is inside the brackets according to the Order of Operations.
::这是嵌入括号的例子, 如上文讨论过的。首先简化括号中的括号。然后根据操作顺序简化括号中的内容。

\begin{aligned} 8 + [4^{2} - 6 \div (5 + 1)] \\ 8 + [4^{2} - 6 \div 6] \\ 8 + [16 - 6 \div 6] \\ 8 + [16 - 1] \\ 8 + 15 \\ 23 \end{aligned}

Review
::回顾

Determine which algebraic property is being used below.
::确定下面使用哪个代数属性。

$\begin{aligned} 8 + 5 = 5 + 8 \end{aligned}$

$\begin{aligned} 7 (x - 2) = 7 x - 14 \end{aligned}$
::7(x-2)=7x-14
$\begin{aligned} \frac{2}{3} \cdot \frac{3}{2} = 1 \end{aligned}$
$\begin{aligned} 4 \cdot (5 \cdot 2) = (4 \cdot 5) \cdot 2 \end{aligned}$
$\begin{aligned} - \frac{1}{4} + 0 = - \frac{1}{4} \end{aligned}$
$\begin{aligned} - 6 + 6 = 0 \end{aligned}$

What is the additive inverse of 1?
::添加剂倒数1是多少?

What is the multiplicative inverse of $\begin{aligned} - \frac{1}{5} \end{aligned}$ ?
::15年的多重反射是什么?

Simplify $\begin{aligned} 6 (4 - 9 + 5) \end{aligned}$ using:
1. the Distributive Property
  ::分配财产账户
2. the Order of Operations
  ::行动命令
3. Do you get the same answer? Why do you think that is?
  ::你得到同样的答案吗?
::简化 6( 4- 9+5 ) : 分配财产行动命令。您是否得到相同的答案 ? 您为什么认为答案相同 ?

Simplify the following expressions using the Order of Operations.
::使用操作命令简化以下表达式。

$\begin{aligned} 12 \div 4 + 3^{3} \cdot 2 - 10 \end{aligned}$
$\begin{aligned} 8 \div 4 + (15 \div 3 - 2^{2}) \cdot 6 \end{aligned}$
$\begin{aligned} \frac{10 - 4 \div 2}{7 \cdot 2 + 2} \end{aligned}$
$\begin{aligned} \frac{1 + 20 - 16 \div 4^{2}}{(5 - 3)^{2} + 12 \div 2} \end{aligned}$
$\begin{aligned} [3 + (4 + 7 \cdot 3) \div 5]^{2} - 47 \end{aligned}$
$\begin{aligned} \frac{7 \cdot 4 - 4}{\frac{6}{2} + 5} \cdot 4^{2} - 18 \end{aligned}$
$\begin{aligned} 6^{2} - [9 + (7 - 5)^{3}] + 49 \div 7 \end{aligned}$
$\begin{aligned} \frac{27 \div 3^{2} + (6 - 2^{2})}{(32 \div 8 + 1) \cdot 3} \end{aligned}$
$\begin{aligned} 6 + 5 \cdot 2 - 9 \div 3 + 4 \end{aligned}$

Using #18 above, insert parenthesis to make the expression equal 1. You may need to use more than one set of parenthesis.
::使用上面#18, 插入括号使表达式等于 1 。您可能需要使用不止一组括号。

Using #18 above, insert parenthesis to make the expression equal 23. You may need to use more than one set of parenthesis.
::使用上面#18, 插入括号使表达式等于 23 。您可能需要使用不止一组括号。

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。

Section outline

General

代数属性

Algebra Properties ::代数属性

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)