Course: 7.4 简化单一变量表达式的总数或差异

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简化单一变量表达式的求和或差异
Marc is heading to his local ice cream store. This particular store doesn't have a lot of flavor choices, but they have the BEST vanilla ice cream around. Marc has taken orders from several of his neighbors, too, and has written them all down so that he can keep track of who wants what.
::马克正前往他所在的冰淇淋店。这家店没有太多的口味选择,但那里有最好的香草冰淇淋。马克也接受了他几个邻居的命令,并把它们都写下来,以便他能够追踪谁想要什么。

The Johnsons - two vanilla ice cream cones
::强生两杯香草冰淇淋甜甜圈

The Mumfords - three vanilla ice cream cones
::Mumfords - 三个香草冰淇淋甜甜甜圈

Jill Stales - one vanilla ice cream cone
::Jill Stales - 一个香草冰淇淋甜筒

The Porters - three vanilla ice cream cones
::波特三个香草冰淇淋甜甜圈

How can Marc write this information as an algebraic expression and then simplify it?
::马克如何将这些信息写成代数表达式然后简化?

In this concept, you will learn how to simplify single variable expressions.
::在此概念中,您将学习如何简化单一变量表达式。

Simplifying Sums or Differences of Single Variable Expressions
::简化单一变量表达式的校验总和或差异

If an expression has only numbers, you can calculate its numerical value. However, if an expression includes variables, it is helpful to simplify the expression.
::如果一个表达式只有数字,您可以计算其数字值。但是,如果一个表达式包含变量,简化表达式会有所帮助。

Look at this example.
::看看这个例子。

Simplify the expression $\begin{align*}6a + 3a\end{align*}$ .
::简化表达式 6a+3a 。

When adding expressions with variables, it is important to remember that only like terms can be combined. For example, $\begin{align*}6a\end{align*}$ and $\begin{align*}3a\end{align*}$ are like terms because both terms include the variable $\begin{align*}a\end{align*}$ . So, you can combine them.
::在添加变量的表达式时,必须记住,只有相似的术语才能合并。例如,6a和3a是相似的术语,因为两个术语都包含变量a。所以,你可以把它们合并在一起。

$\begin{aligned} \begin{array}{rcl} 6 a + 3 a \\ 9 a \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} && 6a+3a \\ && 9a \end{array}\end{align*}$
::6a+3a9a 6a+3a9a

Here is another example.
::下面是另一个例子。

Simplify $\begin{align*}6a + 3\end{align*}$ .
::简化 6a+3。

$\begin{align*}6a\end{align*}$ and $\begin{align*}3\end{align*}$ are not like terms because only one term includes the variable $\begin{align*}a\end{align*}$ . So, you cannot combine them. The expression $\begin{align*}6a + 3\end{align*}$ cannot be simplified any further.
::6a 和 3 与术语不同, 因为只有一个术语包含变量a。所以, 您不能将其合并。表达式 6a+3 无法进一步简化。

Here is another example.
::下面是另一个例子。

Simplify $\begin{align*}15d -2d\end{align*}$ .
::简化15d-2d。

Since $\begin{align*}15d\end{align*}$ and $\begin{align*}2d\end{align*}$ both have the same variable, they are like terms. To find the difference , subtract the numerical parts of the terms the same way you would subtract any numbers.
::由于 15d 和 2d 都有相同的变量, 它们就像条件一样。要找到差异, 请以相同的方式减去条件的数值部分, 您也可以减去任何数字。

$\begin{aligned} 15 d - 2 d = 13 d \end{aligned}$
$\begin{align*}15d - 2d = 13d\end{align*}$
::15天-2天=13天d

The answer is $\begin{align*}13d\end{align*}$ .
::答案是13D

Here is an example using decimals.
::下面是使用小数的示例。

Simplify $\begin{align*}0.4x + 1.3x\end{align*}$ .
::简化 0.4x+1.3x。

Since $\begin{align*}0.4x\end{align*}$ and $\begin{align*}1.3x\end{align*}$ both have the same variable, they are like terms. To find the sum, add the numerical parts of the terms the same way you would add any decimals.
::由于 0.4x 和 1.3x 都有相同的变量, 它们就像条件一样。要找到总和, 请以相同的方式添加术语的数值部分, 并添加任何小数。

$\begin{aligned} 0.4 x + 1.3 x = 1.7 x \end{aligned}$
$\begin{align*}0.4x + 1.3x = 1.7x\end{align*}$
::0.4x+1.3x=1.7x

The answer is $\begin{align*}1.7x\end{align*}$ .
::答案是1.7x。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Marc and all of his ice cream orders.
::之前,你得到一个问题关于马克和他所有的冰淇淋订单。

Marc needs to write an expression to simplify his order : two vanillas, three vanillas, one vanilla, and three vanillas.
::马克需要写一个表达方式来简化他的订单: 两份香草, 三份香草, 一份香草, 三份香草。

Let $\begin{align*}v\end{align*}$ represent an ice cream cone. Then you can represent this situation as a single variable expression .
::让 v 代表一个冰淇淋锥。然后您可以将这种情况作为单一变量表达式来表示。

$\begin{aligned} 2 v + 3 v + v + 3 v \end{aligned}$
$\begin{align*}2v + 3v + v + 3v\end{align*}$
::2v+3v+3v+3v

Looking at this expression, you will see that the variables are all the same. Therefore , simply add the numerical part of each term.
::查看此表达式时, 你会看到变量都是一样的。因此, 只需添加每个词的数值部分。

$\begin{aligned} 2 v + 3 v + v + 3 v = 9 v \end{aligned}$
$\begin{align*}2v + 3v + v + 3v = 9v\end{align*}$

::2v+3v+v+3v=9v

The answer is $\begin{align*}9v\end{align*}$ .
::答案是 9v。

Example 2
::例2

Simplify the expression.
::简化表达式。

$\begin{aligned} 5 a + 4 a - 2 a + 6 a \end{aligned}$
$\begin{align*}5a + 4a- 2a + 6a\end{align*}$
::5a+4a-2a+6a

To simplify this expression, follow the and combine like terms in order from left to right. Here is what the expression looks like after the first two terms have been combined.
::要简化此表达式, 请遵循类似术语的组合, 从左到右顺序排列。这是前两个术语合并后的表达式的外观。

$\begin{aligned} 9 a - 2 a + 6 a \end{aligned}$
$\begin{align*}9a- 2a + 6a\end{align*}$
::9a-2a+6a

Next, perform the subtraction to get: $\begin{align*}7a + 6a\end{align*}$
::下一步,执行减法以获得:7a+6a

Finally, add the terms.
::最后,加上这些术语。

$\begin{aligned} 7 a + 6 a = 13 a \end{aligned}$
$\begin{align*}7a + 6a=13a\end{align*}$
::7a+6a=13a

The answer is $\begin{align*}13a\end{align*}$ .
::答案是13a。

Simplify each sum or difference when possible.
::尽可能简化每项总和或差额。

Example 3
::例3

$\begin{aligned} 3 a + 12 a \end{aligned}$
$\begin{align*}3a + 12a\end{align*}$
::3a+12a 3a+12a

These are like terms, so add the numerical parts together.
::这些都是相似的术语,所以将数字部分加在一起。

The answer is $\begin{align*}15a\end{align*}$ .
::答案是15a。

Example 4
::例4

$\begin{aligned} 16 x - 2 x \end{aligned}$
$\begin{align*}16x- 2x\end{align*}$
::16 - 2x 16 - 2x

These are like terms, so subtract the numerical parts.
::这些都是相似的术语,所以减去数字部分。

The answer is $\begin{align*}14x\end{align*}$ .
::答案是14x。

Example 5
::例5

$\begin{aligned} 7 y + 2 x \end{aligned}$
$\begin{align*}7y + 2x\end{align*}$
::7y+2x 7y+2x

These are not like terms.
::这和条件不一样

The terms are not alike so you cannot combine them. The expression is in the simplest form already.
::术语不相似, 所以您无法将其合并。表达式已经以最简单的形式出现。

The answer is $\begin{align*}7y+2x\end{align*}$ .
::答案是7y+2x。

Review
::回顾

Simplify each sum or difference by combining like terms.
::将类似术语合并,简化每项总和或差额。

$\begin{align*}6a + 7a\end{align*}$
::6a+7a 6a+7a

$\begin{align*}7x-2x\end{align*}$
::7x-2x

$\begin{align*}6y + 12y\end{align*}$
::6y+12y

$\begin{align*}8a + 12a\end{align*}$
::8a+12a

$\begin{align*}12y-7y\end{align*}$
::12-7y

$\begin{align*}8a + 15a\end{align*}$
::8a+15a

$\begin{align*}13b- 9b\end{align*}$
::13b-9b

$\begin{align*}22x + 19x\end{align*}$
::22x19xx 22x+19x

$\begin{align*}45y-12y\end{align*}$
::45y-12y

$\begin{align*}16a + 18a + 9a\end{align*}$
::16a+18a+9a

$\begin{align*}14x-6x + 2x\end{align*}$
::14 - 6x+2x

$\begin{align*}21a + 14a-15a\end{align*}$
::21a+14a-15a

$\begin{align*}33b + 13b + 8b\end{align*}$
::33b+13b+8b

$\begin{align*}45x + 67x- 29x\end{align*}$
::45x+67x-29x

$\begin{align*}92y + 6y-54y\end{align*}$
::92y+6y-54y

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

简化单一变量表达式的求和或差异

Simplifying Sums or Differences of Single Variable Expressions ::简化单一变量表达式的校验总和或差异

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 Sums or Differences of Single Variable Expressions
::简化单一变量表达式的校验总和或差异

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