Course: 7.6 简化涉及多重操作的变量表达式

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简化涉及多个操作的变量表达式
A farmer, Kelly, has two lovely plots of rectangular land to grow some vegetables. Her friend will help her plant in the spring. One plot of land is $\begin{aligned} 8 a \end{aligned}$ by $\begin{aligned} 6 a \end{aligned}$ and the other plot of land is $\begin{aligned} 3 a \end{aligned}$ by $\begin{aligned} 4 a \end{aligned}$ . The two plots of land are going to be combined so they can grow more vegetables. How can Kelly find the total area for both plots of farmable land?
::农民凯利拥有两块可爱的长方形土地来种植蔬菜。她的朋友会在春天帮助她种植。一块土地是8a乘6a,另一块土地是3a乘4a。两块土地将合并,这样他们就能种植更多的蔬菜。凯利如何为这两块可耕地找到总面积?

In this concept, you will learn to simplify variable expressions involving multiple operations .
::在此概念中,您将学会简化涉及多个操作的变量表达式。

Simplifying Variable Expressions Involving Multiple Operations
::简化涉及多个操作的变量表达式

Sometimes, you may need to simplify algebraic expressions that involve more than one operation . Use what you know about simplifying sums, differences, products, or quotients of algebraic expressions to help you do this.
::有时,您可能需要简化涉及多个操作的代数表达式。使用您所知道的简化数值、差异、产品或代数表达式的商数来帮助您做到这一点。

When evaluating expressions, it is important to keep in mind the , which is
::在评价表达式时,必须铭记

First, do the computation inside " data-term="Parentheses" role="term" tabindex="0"> parentheses .
::首先,在括号内进行计算。

Second, evaluate any exponents.
::第二,评估任何前台。

Third, multiply and divide in order from left to right.
::第三,从左到右的成倍和分化。

Finally, add and subtract in order from left to right.
::最后,按顺序从左向右增减。

Now let’s look at an example.
::现在让我们来举一个例子。

Simplify this expression $\begin{aligned} 7 n + 8 n \cdot 3 \end{aligned}$
::简化此表达式 7n+8n}3

First, simplify according to the order of operations. According to the order of operations, you should multiply first.
::首先,根据操作顺序简化。根据操作顺序,您应该先乘。

$\begin{aligned} 7 n + 8 n \cdot 3 = 7 n + 24 n . \end{aligned}$

::7n+8n3=7n+24n.

Next, add like terms .
::之后加上类似术语。

$\begin{aligned} 7 n + 24 n = 31 n \end{aligned}$

::7n+24n=31n

The answer is $\begin{aligned} 31 n \end{aligned}$ .
::答案是31n

Here is another example.
::下面是另一个例子。

Simplify the expression $\begin{aligned} 10 p - 7 p + 8 p \div 2 p \end{aligned}$ .
::简化表达式 10p-7p+8p2p 。

First, follow the order of operations and rewrite the division as a fraction .
::首先,按照操作顺序,将分数重写为分数。

$\begin{aligned} \frac{8 p}{2 p} \end{aligned}$

::8p2p8p2p

Next, simplify the fraction, assuming $\begin{aligned} p \end{aligned}$ is not equal to zero.
::其次,简化分数,假设p不等于零。

$\begin{aligned} \frac{8 p}{2 p} = \frac{8}{2} \times \frac{p}{p} = 4 \times 1 = 4 \end{aligned}$

::8p2=82xpp=4x1=4

Next, rewrite the equation .
::下一位,重写方程式

$\begin{aligned} 10 p - 7 p + \frac{8 p}{2 p} = 10 p - 7 p + 4 \end{aligned}$

::10p-7p+8p2p=10p-7p+4

Simplify, by combing like terms.
::简化,像用词一样梳理。

$\begin{aligned} 10 p - 7 p + 4 = 3 p + 4 \end{aligned}$

::10p-7p+4=3p+4

The answer is $\begin{aligned} 3 p + 4 \end{aligned}$ .
::答案是3p+4。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Kelly and her two plots of land.
::早些时候,你得到一个问题凯莉和她两块土地。

One is $\begin{aligned} 8 a \end{aligned}$ by $\begin{aligned} 6 a \end{aligned}$ and the other is $\begin{aligned} 3 a \end{aligned}$ by $\begin{aligned} 4 a \end{aligned}$ .
::一个是8a乘6a,另一个是3a乘4a。

The plots of land need to be combined to find the total area of farmable land.
::土地需要合并,以找到可耕地的总面积。

First, consider the equation for the area of a rectangle.
::首先,考虑矩形区域的方程。

$\begin{aligned} Area of a rectangle = length \times width \end{aligned}$

::矩形区域=长xwidth

Next, calculate the area of the first rectangular plot of land.
::接下来,计算第一块长方形土地的面积。

$\begin{aligned} 8 a \times 6 a = 48 a^{2} \end{aligned}$

::8ax6a=48a2

Then, calculate the area of the second rectangular plot of land.
::然后计算第二个长方形土地的面积。

$\begin{aligned} 3 a \times 4 a = 12 a^{2} \end{aligned}$

::3ax4a=12a2

Next, write and expression for the total area of the two plots of land.
::接下来是两块土地总面积的写作和表达。

$\begin{aligned} 48 a^{2} + 12 a^{2} \end{aligned}$

::48a2+12a2

Finally combine like terms.
::最后,将类似术语结合起来。

$\begin{aligned} 48 a^{2} + 12 a^{2} = 60 a^{2} \end{aligned}$

::48a2+12a2=60a2

The answer is $\begin{aligned} 60 a^{2} \end{aligned}$ .
::答案是60a2

Example 2
::例2

Samera has twice as many pets as Amit has. Kyra has 4 times as many pets as Amit has. Let $\begin{aligned} a \end{aligned}$ represent the number of pets Amit has.
::Samera的宠物数量是Amit的两倍。 Kyra的宠物数量是Amit的4倍。请代表Amit的宠物数量。

Write an expression to represent the number of pets Samera has.
::写一个表达式来代表Samera的宠物数量。

Write an expression to represent the number of pets Kyra has.
::写一个表达式来代表 Kyra 的宠物数量。

Write an expression to represent the number of pets Samera and Kyra have all together.
::写一个表达式来代表Samera和Kyra的宠物数量。

First, answer part a .
::第一,回答第一部分。

Samera has twice as many pets as Amit. Since Amit has a pets, Samera has $\begin{aligned} 2 a \end{aligned}$ pets.
::Samera养的宠物是Amit的两倍。自从Amit养了宠物,Samera养了2只宠物。

Next, answer part b .
::接下去回答B部分。

Kyra has 4 times as many pets as Amit has. Since Amit has a pets, Kyra has $\begin{aligned} 4 a \end{aligned}$ pets.
::Kyra养的宠物是Amit的4倍。由于Amit养的宠物,Kyra养的宠物是4A。

Finally, answer part c .
::最后,回答C部分。

To find the number of pets Samera and Kyra have “all together,” write an addition expression.
::为了找到Samera和Kyra的宠物数量,

$\begin{aligned} 2 a + 4 a \end{aligned}$

::2a+4a

Finally, combine like terms.
::最后, 合并起来像术语一样。

$\begin{aligned} 2 a + 4 a = 6 a \end{aligned}$

::2a+4a=6a

The answer is $\begin{aligned} 6 a \end{aligned}$ .
::答案是6a。

Simplify each expression.
::简化每个表达式。

Example 3
::例3

$\begin{aligned} 4 a + 9 a - 7 \end{aligned}$

::4a+9a-7

Combine the like terms $\begin{aligned} 4 a \end{aligned}$ and $\begin{aligned} 9 a \end{aligned}$ .
::结合4a和9a的术语

The answer is $\begin{aligned} 13 a - 7 \end{aligned}$ .
::答案是13a-7。

Example 4
::例4

$\begin{aligned} \frac{14 x}{2} + 9 x \end{aligned}$

::14x2+9x 14x2+9x

First, follow the order of operations and do the division first.
::首先,按照行动顺序,先做分工。

$\begin{aligned} \frac{14 x}{2} + 9 x = 7 x + 9 x \end{aligned}$

::14x2+9x=7x+9x

Next, combine like terms.
::接下来,把术语合并起来。

$\begin{aligned} 7 x + 9 x = 16 x \end{aligned}$

::7x+9x=16x

The answer is $\begin{aligned} 16 x \end{aligned}$ .
::答案是16x。

Example 5
::例5

$\begin{aligned} 6 b - 2 b + 5 b - 8 \end{aligned}$

::6b-2b+5b-8

Follow the order of operations and perform the addition and subtraction from left to right.
::遵循操作顺序,执行从左向右增减。

$\begin{aligned} 6 b - 2 b + 5 b - 8 = 9 b - 8 \end{aligned}$

::6b-2b+5b-8=9b-8

The answer is $\begin{aligned} 9 b - 8 \end{aligned}$ .
::答案是9b-8。

Review
::回顾

Simplify each expression involving multiple operations.
::简化涉及多个操作的每个表达式。

$\begin{aligned} 6 a + 4 a - 2 b \end{aligned}$
::6a+4a-2b

$\begin{aligned} 16 b - 4 b \cdot 2 \end{aligned}$
::16b-4b2

$\begin{aligned} 22 a \div 2 + 14 a \end{aligned}$
::22a2+14a

$\begin{aligned} 19 x - 5 x \cdot 2 \end{aligned}$
::19 - 5x% 2

$\begin{aligned} 16 y - 12 y \div 2 \end{aligned}$
::16-12y2

$\begin{aligned} 16 a - 4 a - 12 b \end{aligned}$
::16a - 4a - 12b

$\begin{aligned} 26 a + 14 a + 12 b + 2 b \end{aligned}$
::26a+14a+12b+2b

$\begin{aligned} 36 a + 4 a - 2 b + 5 b \end{aligned}$
::36a+4a-2b+5b

$\begin{aligned} 18 a + 4 a + 12 y \end{aligned}$
::18a+4a+12y

$\begin{aligned} 46 a + 34 a - 12 b + 14 b \end{aligned}$
::46a+34a-12b+14b

$\begin{aligned} 16 y + 4 y - 2 x \end{aligned}$
::16y+4y-2x 16y+4y-2x

$\begin{aligned} 6 x + 4 x + 2 x + 4 y - 19 z \end{aligned}$
::6x+4x+2x+4y-19z

$\begin{aligned} 26 y - 12 y \div 2 \end{aligned}$
::26-12y2

$\begin{aligned} 36 y - 12 y \div 12 \end{aligned}$
::36-12y12

$\begin{aligned} 46 y + 12 y \div 2 \end{aligned}$
::46+12y @%2

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

简化涉及多个操作的变量表达式

Simplifying Variable Expressions Involving Multiple Operations ::简化涉及多个操作的变量表达式

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)

Resources ::资源

Simplifying Variable Expressions Involving Multiple Operations
::简化涉及多个操作的变量表达式

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)

Resources
::资源