Course: 1.9带基本运算的表达式求值

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带基本运算的表达式求值
School is about to start back up again and Lucy is at the store buying supplies. In her basket she has 5 notebooks for $2 each, 2 packs of pencils for $4 each, and 2 packs of pens for $5 each. She also picked out a big binder for $12. Luckily, she has a coupon for $3 off the binder. How can Lucy write and evaluate a numerical expression to calculate the total amount she will spend on her school supplies?
::学校即将重新开始,露西正在商店购买用品。在她的篮子里,她有5本笔记本,每本2美元,两包铅笔,每本4美元,两包笔,每本5美元。她还挑选了一个大装饰品,12美元。幸运的是,她有一个3美元的优惠券。露西如何写和评估一个数字表达方式来计算她将花在学校用品上的总额?

In this concept, you will learn how to evaluate variable expressions involving the four basic operations .
::在此概念中,您将学会如何评价涉及四个基本操作的变量表达式。

Evaluating Expressions with Basic Operations
::使用基本操作评价表达式

An expression is a mathematical phrase that contains numbers and operations. A numerical expression is an expression with only numbers and operations and no variables. A variable expression is an expression that contains variables.
::表达式是一个包含数字和操作的数学短语。数字表达式是一个仅包含数字和操作且没有变量的表达式。变量表达式是一个包含变量的表达式。

To evaluate a variable expression means to find the value of the expression for given values of the variables. To evaluate, substitute the given values for the variables in the expression and simplify using the (PEMDAS).
::要评价变量表达式, 需要为变量的指定值查找表达式的值。要评估, 请将给定值替换为表达式中的变量, 并使用( PEMDAS) 简化。

Remember that the order of operations are:
::记住行动顺序是:

P arentheses: Start by simplifying any part of the expression in " data-term="Parentheses" role="term" tabindex="0"> parentheses using the order of operations.
::括号:从使用操作顺序简化括号中的任何部分开始。

E xponents: Rewrite any terms that contain exponents without exponents.
::指数:重写任何含有引言但无引言的术语。

M ultiplication/ D ivision: Do any multiplication and/or division in order from left to right.
::倍数/司:从左到右做任何倍数和(或)除法。

A ddition/ S ubtraction: Do any addition and/or subtraction in order from left to right.
::加/减:按从左到右顺序增加和(或)减。

The acronym PEMDAS can help you to remember the order of operations. Just remember that multiplication and division go together and addition and subtraction go together.
::缩略语 PEMDAS 可以帮助您记住操作的顺序。只要记住乘法和除法是一起的, 加法和减法是一起的。

Here is an example.
::举一个例子。

Evaluate the expression $\begin{aligned} 4 b + 12 \div 4 - 8 \end{aligned}$ if $\begin{aligned} b = 6 \end{aligned}$ .
::如果b = 6,评价4 b+ 12 = 4 = 8 = 6 = 4 b + 12 = 4 = 8。

First, substitute 6 in for the $\begin{aligned} b \end{aligned}$ in the expression.
::首先,用B的6取代B的表达式。

$\begin{aligned} 4 (6) + 12 \div 4 - 8 \end{aligned}$

Next, notice that this expression has multiplication, addition, division, and subtraction. Remember that $\begin{aligned} 4 (6) \end{aligned}$ is multiplication. Start by doing the multiplication and division from left to right. First you will multiply $\begin{aligned} 4 (6) \end{aligned}$ .
::下一步请注意, 此表达式有乘法、添加、除法和减法。记住 4 (6) 是乘法。从乘法和分法从左向右开始。首先, 您会乘法 4 (6) 。

$\begin{aligned} 4 (6) + 12 \div 4 - 8 = 24 + 12 \div 4 - 8 \end{aligned}$

Next, divide $\begin{aligned} 12 \div 4 \end{aligned}$ .
::下个分数是12+4

$\begin{aligned} 24 + 12 \div 4 - 8 = 24 + 3 - 8 \end{aligned}$

Now, add and subtract from left to right. First, add $\begin{aligned} 24 + 3 \end{aligned}$ .
::现在,从左向右增减。首先,增加24+3。

$\begin{aligned} 24 + 3 - 8 = 27 - 8 \end{aligned}$

Finally, subtract.
::最后,减去。

$\begin{aligned} 27 - 8 = 19 \end{aligned}$

The answer is 19.
::答案是19个

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Lucy, who is shopping for school supplies.
::早些时候,你被问及露西的问题, 露西正在购物购买学校用品。

She has 5 notebooks for $2 each, 2 packs of pencils for $4 each, 2 packs of pens for $5 each, and a big binder for $12. She also has a coupon for $3 off the binder. Lucy wants to know how much she will be spending on supplies.
::她有5本笔记本,每本2美元,两包铅笔,每本4美元,两包笔,每本5美元,还有一卷大卷装饰品,12美元。她还有一张3美元的优惠券。露西想知道她在用品上要花多少钱。

First, write an expression for each item that represents how much she is spending on that item.
::首先,为每个项目写一个表达式,表示她在该项目上花费了多少钱。

Notebooks: $\begin{aligned} 5 \times $ 2 \end{aligned}$
::笔记本:5x2美元

Pencils: $\begin{aligned} 2 \times $ 4 \end{aligned}$
::铅笔: 2 × 美元 4

Pens: $\begin{aligned} 2 \times $ 5 \end{aligned}$
::笔数: 2 × 5美元

Binder: $\begin{aligned} $ 12 - $ 3 \end{aligned}$
::BING: 12美元 - 3美元

Now, add everything together to create one expression that represents the total amount she is spending.
::现在,将所有东西加在一起来创建一个表达式代表她所花费的总金额。

$\begin{aligned} 5 \times 2 + 2 \times 4 + 2 \times 5 + 12 - 3 \end{aligned}$

Next, notice that this expression has multiplication, addition, and subtraction. Start by doing the multiplication from left to right. First you will multiply $\begin{aligned} 5 \times 2 \end{aligned}$ . Then continue multiplying from left to right.
::接下来请注意, 此表达式有乘法、加法和减法。首先从左向右进行乘法。首先, 您会乘法 5 x 2 。然后继续从左向右乘法。

$\begin{aligned} \begin{array}{rcl} 5 \times 2 + 2 \times 4 + 2 \times 5 + 12 - 3 & = & 10 + 2 \times 4 + 2 \times 5 + 12 - 3 \\ = & 10 + 8 + 2 \times 5 + 12 - 3 \\ = & 10 + 8 + 10 + 12 - 3 \end{array} \end{aligned}$

Next, add and subtract from left to right.
::下一步,从左向右增减。

$\begin{aligned} \begin{array}{rcl} 10 + 8 + 10 + 12 - 3 & = & 18 + 10 + 12 - 3 \\ = & 28 + 12 - 3 \\ = & 40 - 3 \\ = & 37 \end{array} \end{aligned}$

The answer is that Lucy is spending $37 on school supplies.
::答案是露西花37美元购买学校用品

Example 2
::例2

Evaluate $\begin{aligned} 6 x + 36 \div 4 - 3 + 8 \end{aligned}$ if $\begin{aligned} x \end{aligned}$ is 4.
::评价 6 x + 36 + 4 - 3 + 8 如果 x 是 4 ,则评价 6 x + 36 + + 4 - 3 + 8 。

First, substitute 4 in for the $\begin{aligned} x \end{aligned}$ in the expression.
::首先,将X改为4。

$\begin{aligned} 6 (4) + 36 \div 4 - 3 + 8 \end{aligned}$

Next, notice that this expression has multiplication, addition, division, and subtraction. Start by doing the multiplication and division from left to right. First you will multiply $\begin{aligned} 6 (4) \end{aligned}$ .
::下一步请注意, 此表达式有乘法、添加、分割和减法。从进行乘法和分法从左向右开始。首先, 您会乘法 6 (4 ) 。

$\begin{aligned} 6 (4) + 36 \div 4 - 3 + 8 = 24 + 36 \div 4 - 3 + 8 \end{aligned}$

Next, divide $\begin{aligned} 36 \div 4 \end{aligned}$ .
::下一个是36+4

$\begin{aligned} 24 + 36 \div 4 - 3 + 8 = 24 + 9 - 3 + 8 \end{aligned}$

Now, add and subtract from left to right. First, add $\begin{aligned} 24 + 9 \end{aligned}$ .
::现在,从左向右增减。首先,增加24+9。

$\begin{aligned} 24 + 9 - 3 + 8 = 33 - 3 + 8 \end{aligned}$

Then, subtract $\begin{aligned} 33 - 3 \end{aligned}$ .
::然后,减去33 - 3。

$\begin{aligned} 33 - 3 + 8 = 30 + 8 \end{aligned}$

Finally, add.
::最后,增加。

$\begin{aligned} 30 + 8 = 38 \end{aligned}$

The answer is 38.
::答案是38岁

Example 3
::例3

Evaluate $\begin{aligned} 5 a + 6 \div 2 + 9 \end{aligned}$ if $\begin{aligned} a = 5 \end{aligned}$ .
::如果a=5,评价5 a+6 +2+9。

First, substitute 5 in for the $\begin{aligned} a \end{aligned}$ in the expression.
::首先,以5取代a的表达式。

$\begin{aligned} 5 (5) + 6 \div 2 + 9 \end{aligned}$

Next, notice that this expression has multiplication, addition, and division. Start by doing the multiplication and division from left to right. First you will multiply $\begin{aligned} 5 (5) \end{aligned}$ . Then divide.
::下一步请注意, 此表达式有乘法、添加和除法。从进行乘法和分法从左向右开始。首先, 您会乘法 5 (5) 。然后分隔。

$\begin{aligned} \begin{array}{rcl} 5 (5) + 6 \div 2 + 9 & = & 25 + 6 \div 2 + 9 \\ = & 25 + 3 + 9 \end{array} \end{aligned}$

Next, add from left to right.
::下一步,从左到右添加。

$\begin{aligned} \begin{array}{rcl} 25 + 3 + 9 & = & 28 + 9 \\ = & 37 \end{array} \end{aligned}$

The answer is 37.
::答案是37岁

Example 4
::例4

Evaluate $\begin{aligned} 7 x + 14 \div 7 - 3 \end{aligned}$ if $\begin{aligned} x = 4 \end{aligned}$ .
::评价 7 x + 14 + 7 - 3 如果x = 4 则评价 7 x + 14 * 7 - 3 。

First, substitute 4 in for the $\begin{aligned} x \end{aligned}$ in the expression.
::首先,将X改为4。

$\begin{aligned} 7 (4) + 14 \div 7 - 3 \end{aligned}$

Next, notice that this expression has multiplication, addition, division, and subtraction. Start by doing the multiplication and division from left to right. First you will multiply $\begin{aligned} 7 (4) \end{aligned}$ . Then divide.
::下一步请注意, 此表达式有乘法、添加、分割和减法。从进行乘法和分法从左向右开始。首先, 您会乘法 7 (4) 。然后分隔。

$\begin{aligned} \begin{array}{rcl} 7 (4) + 14 \div 7 - 3 & = & 28 + 14 \div 7 - 3 \\ = & 28 + 2 - 3 \end{array} \end{aligned}$

Next, add and subtract from left to right. First you will add $\begin{aligned} 28 + 2 \end{aligned}$ . Then subtract.
::下一步,从左向右增减。首先,你将增加28+2。然后减去。

$\begin{aligned} \begin{array}{rcl} 28 + 2 - 3 & = & 30 - 3 \\ = & 27 \end{array} \end{aligned}$

The answer is 27.
::答案是27个

Example 5
::例5

Evaluate $\begin{aligned} \frac{48}{x} + 5 y - 7 \end{aligned}$ if $\begin{aligned} x \end{aligned}$ is 6 and $\begin{aligned} y \end{aligned}$ is 9.
::评价48x+5y-7,如果x6,y为9。

First, substitute 6 in for the $\begin{aligned} x \end{aligned}$ and 9 in for the $\begin{aligned} y \end{aligned}$ in the expression.
::首先,将 X 的 6 替换为 y 的 x 和 9 替换为 y 的表达式。

$\begin{aligned} \frac{48}{6} + 5 (9) - 7 \end{aligned}$

Next, notice that this expression has division, addition, multiplication, and subtraction. Remember that a fraction bar means division so $\begin{aligned} \frac{48}{6} \end{aligned}$ is the same as $\begin{aligned} 48 \div 6 \end{aligned}$ .
::接下来请注意, 此表达式有分割、添加、乘法和减法。记住, 分数条意味着分割, 48 6 和 48 6 相同。

Start by doing the multiplication and division from left to right. First you will divide $\begin{aligned} \frac{48}{6} \end{aligned}$ . Then multiply.
::开始从左向右进行乘法和分法的乘法。首先, 您将分隔 48 6 。然后乘法。

$\begin{aligned} \begin{array}{rcl} \frac{48}{6} + 5 (9) - 7 & = & 8 + 5 (9) - 7 \\ = & 8 + 45 - 7 \end{array} \end{aligned}$

Next, add and subtract from left to right. First you will add $\begin{aligned} 8 + 45 \end{aligned}$ . Then subtract.
::下一步,从左向右增减。首先,你将增加8+45。然后减去。

$\begin{aligned} \begin{array}{rcl} 8 + 45 - 7 & = & 53 - 7 \\ = & 46 \end{array} \end{aligned}$

The answer is 46.
::答案是46个

Review
::回顾

Use the order of operations to evaluate each numerical expression.
::使用操作顺序来评价每个数字表达式。

$\begin{aligned} 5 + 3 \times 4 \end{aligned}$

$\begin{aligned} 6 \times 2 + 5 \times 3 \end{aligned}$

$\begin{aligned} 4 + 5 \times 2 - 9 \end{aligned}$

$\begin{aligned} 4 + 6 \div 2 + 10 - 3 \end{aligned}$

$\begin{aligned} 8 - 15 \div 3 + 4 \times 5 \end{aligned}$

Evaluate the following expressions when $\begin{aligned} x \end{aligned}$ is 4.
::x 是 4 时评价以下表达式:

$\begin{aligned} 2 x + 28 \div 4 - 3 \end{aligned}$
::2 x + 28 + 28 * * 4 - 3

$\begin{aligned} 6 x + 30 \div 2 - 10 \end{aligned}$
::6 x + 30 + 30 * 2 - 10

$\begin{aligned} 5 x + x - 3 + 8 \end{aligned}$
::5 x + x + x - 3 + 8

$\begin{aligned} 3 x + 36 \div 9 - 3 + 1 \end{aligned}$
::3 x + 36 + + 9 - 3 + 1

$\begin{aligned} 5 x + 9 - 3 + 8 \end{aligned}$
::5 x + 9 + 9 - 3 + 8

$\begin{aligned} 4 x + 30 \div 15 - 3 + 8 \end{aligned}$
::4 x + 30 + 30 * 15 - 3 + 8

$\begin{aligned} 8 x + 2 x - 3 + 18 \end{aligned}$
::8 x + 2 x + 3 + 18

$\begin{aligned} 2 x + x \div 2 - 3 + 8 \end{aligned}$
::2 x + x + + + 2 - 3 + 8

$\begin{aligned} 9 x + 3 x \div 2 + 8 \end{aligned}$

::9 x + 3 x + 3 x 2 + 8

$\begin{aligned} 12 x + 4 x \div 4 - 6 + 8 \end{aligned}$
::12 x + 4 x + 4 + + + 4 + + + + + + 4 - 4 - 6 + 8

Review (Answers)
::回顾(答复)

Section outline

General

带基本运算的表达式求值

Evaluating Expressions with Basic Operations ::使用基本操作评价表达式

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)

Evaluating Expressions with Basic Operations
::使用基本操作评价表达式

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)