Section: 单变量表达式的和与区别 | 12.4单变量表达式的和与区别

Section outline

Jacob’s mother has put him in charge of planning a surprise party for his little brother, Kyle. Jacob plans to serve cake and ice-cream cups. He does not know how much each ice-cream cup will cost. He has invited 24 of Kyle’s friends and 6 other family members. Jacob needs to figure out how much the ice-cream cups will cost so he can get the money from his mother. How can he formulate a single variable expression to determine the total costs of the ice-cream cups?
::Jacob的母亲让他负责为他弟弟Kyle策划一个惊喜派对。Jacob打算提供蛋糕和冰淇淋杯。他不知道每个冰淇淋杯要花多少钱。他邀请了24位Kyle的朋友和6位其他家庭成员。Jacob需要弄清楚冰淇淋杯要花多少钱才能从他母亲那里拿到钱。他如何用一个变量表达来决定冰淇淋杯的总成本?

In this concept, you will learn to simplify single- variable expressions that include sums and differences.
::在此概念中,您将学会简化包含数字和差异的单一变量表达式。

Simplifying Expressions Involving Sums and Differences
::涉及总和和差异的简化表达式

In mathematics, simplifying does not mean solving something. Simplifying means making something smaller.
::在数学中,简化并不意味着解决什么问题。简化意味着制造更小的东西。

Sometimes, you will be given an expression using variables where there is more than one term . A term is a number with a variable. Here is an example of a term.
::有时,您会使用多个术语的变量获得一个表达式。一个术语是带有变量的数字。这里是术语示例。

$\begin{aligned} 4 x \end{aligned}$
::4 x 4 x

You have not been given a value for $\begin{aligned} x \end{aligned}$ , so this term cannot be simplified. If you had been given a value for $\begin{aligned} x \end{aligned}$ , then you could evaluate the expression.
::您没有给 x 给定值, 因此此术语无法简化。如果您给定了 x 的值, 那么您可以对表达式进行评价。

When there is more than one like term in an expression, you can simplify the expression. A like term means that the terms in question use the same variable.
::当表达式中有多个类似术语时,您可以简化表达式。类似术语意味着相关术语使用相同的变量。

$\begin{aligned} \underline{4 x} \end{aligned}$ and $\begin{aligned} \underline{5 x} \end{aligned}$ are like terms . They both have $\begin{aligned} \underline{x} \end{aligned}$ as the variable. They are alike.
::4 x _ 和 5 x _ 类似术语。它们都有 x _ 作为变量。它们相似。

$\begin{aligned} \underline{6 x} \end{aligned}$ and $\begin{aligned} \underline{2 y} \end{aligned}$ are not like terms. One has an $\begin{aligned} \underline{x} \end{aligned}$ and one has a $\begin{aligned} \underline{y} \end{aligned}$ . They are not alike.
::6 x _ 和 2 y y 和 6 x _ 和 6 x 和 6 y 和 6 的术语不同。一个人有一个 x _ , 一个人有一个 y 。它们不相似。

Expressions with like terms can be simplified.
::类似术语的表达式可以简化。

You can also simplify the sums and differences of expressions with like terms. Let’s start with sums.
::您也可以用类似术语简化表达方式的大小和差别。让我们从数字开始。

Here is an expression.
::这是一种表达方式。

$\begin{aligned} 5 x + 7 x \end{aligned}$

First, look to see if these terms are alike. Both of them have an $\begin{aligned} x \end{aligned}$ , so they are alike.
::首先,看看这两个词是否相似。两个词都有一个x,所以它们都一样。

Next, simplify them by adding the numerical part of the terms together. The $\begin{aligned} x \end{aligned}$ stays the same.
::接下来,通过将术语的数值部分加在一起来简化它们。 x 保持不变。

$\begin{aligned} \begin{array}{rcl} 5 x + 7 x \\ 5 + 7 = 12 \\ So, & 5 x + 7 x = 12 x \end{array} \end{aligned}$

You can think of the $\begin{aligned} x \end{aligned}$ as a label that lets you know that the terms are alike.
::您可以将 x 视为一个标签, 让你知道这些术语是相似的。

Here is another expression.
::下面是另一种表达方式。

$\begin{aligned} 7 x + 2 x + 5 y \end{aligned}$

First, look to see if the terms are alike.
::首先,看看这些条件是否相似。

Two of the terms have $\begin{aligned} x \end{aligned}$ ‘s and one has a $\begin{aligned} y \end{aligned}$ . The two with the $\begin{aligned} x \end{aligned}$ ‘s are alike. The one with the $\begin{aligned} y \end{aligned}$ is not alike. You can simplify the ones with the $\begin{aligned} x \end{aligned}$ ‘s by adding the numerical part of the terms.
::其中两个词有x ' s, 一个词有一个 y 。带有 x ' s 的这两个词是相同的。有 y 的则不一样。您可以通过添加术语的数值部分来简化有 x ' s 的词。

Next, simplify the like terms.
::接下来,简化类似术语。

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

Then, because $\begin{aligned} 5 y \end{aligned}$ can’t be simplified, it stays the same.
::因为5人无法简化,

The answer is $\begin{aligned} 9 x + 5 y \end{aligned}$ .
::答案是 9 x + 5 y 。

You can also simplify expressions with differences and like terms.
::您也可以简化带有差异和类似术语的表达式。

Here is an expression.
::这是一种表达方式。

$\begin{aligned} 9 y - 2 y \end{aligned}$

First, you can see that these terms are alike because they both have $\begin{aligned} y \end{aligned}$ ’s. Simplify the expression by subtracting the numerical part of the terms.
::首先,你可以看到,这些术语是相似的,因为它们都有y。通过减去术语的数值部分来简化表达式。

$\begin{aligned} \begin{array}{rcl} 9 - 2 & = & 7 \\ So, 9 y - 2 y & = & 7 y \end{array} \end{aligned}$

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

Sometimes you can combine like terms that have both sums and differences in the same problem.
::有时,你可以将类似术语结合起来,在同一个问题中,这些术语既有数字也有差异。

Here is another expression.
::下面是另一种表达方式。

$\begin{aligned} 8 x - 3 x + 2 y + 4 y \end{aligned}$

First, group and simplify the like terms.
::首先,组和简化类似术语。

$\begin{aligned} \begin{array}{rcl} 8 x - 3 x = 5 x \\ 2 y + 4 y = 6 y \end{array} \end{aligned}$

Next, put it all together.

The answer is $\begin{aligned} 5 x + 6 y \end{aligned}$ .
::接下去,把全部放在一起。答案是 5 x + 6 y 。

Remember that you can only combine terms that are alike.
::记住,你只能将相似的术语结合起来。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Kyle’s surprise birthday party.
::更早之前, 凯尔的惊喜生日派对给您带来了一个问题。

Jacob needs to know the cost of 24 ice-cream cups for Kyle’s friends and 6 ice-cream cups for the invited family members.
::Jacob需要知道24个冰淇淋杯给Kyle的朋友和6个冰淇淋杯给受邀家庭成员的费用。

First, identify the numbers.
::首先,确定数字。

24 and 6
::24和6

Next, make $\begin{aligned} x \end{aligned}$ represent the cost of each ice-cream cup.
::接下来,做x代表每个冰淇淋杯的成本。

$\begin{aligned} x \end{aligned}$
::x x

Then, identify the key word “and” which means addition .
::然后,确定关键词“和”,这意味着添加。

$\begin{aligned} + \end{aligned}$

Now, write an expression representing the situation.
::现在,写一个表达式代表情况。

$\begin{aligned} 24 x + 6 x \end{aligned}$

Finally, simplify the expression by adding the numerical value of like terms.
::最后,通过添加类似术语的数值来简化表达式。

$\begin{aligned} \begin{array}{rcl} 24 + 6 & = & 25 \\ So, 24 x + 6 x & = & 30 x \end{array} \end{aligned}$

The answer is $\begin{aligned} 30 x \end{aligned}$ .
::答案是30x。

Jacob will need to multiply the price of an ice-cream cup by 30 to determine the total cost.
::Jacob需要把冰淇淋杯的价格乘以30 才能确定总成本

Example 2
::例2

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} 5 x + 2 x - 1 x + 6 y - 4 y \end{aligned}$

First, combine the like terms.
::首先,将类似条件结合起来。

$\begin{aligned} 5 x + 2 x - 1 x \\ 6 y - 4 y \end{aligned}$

Next, simplify the like terms.
::接下来,简化类似术语。

$\begin{aligned} \begin{array}{rcl} 5 x + 2 x - 1 x & = & 6 x \\ 6 y - 4 y & = & 2 y \end{array} \end{aligned}$

Then, put those simplified terms together.
::然后,把这些简化的术语放在一起。

$\begin{aligned} 6 x + 2 y \end{aligned}$

The answer is $\begin{aligned} 6 x + 2 y \end{aligned}$ .
::答案是 6 x + 2 y 。

Example 3
::例3

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} 7 z + 2 z + 4 z \end{aligned}$

First, look to see if these terms are alike. They all have an $\begin{aligned} z \end{aligned}$ , so they are alike.
::首先,看看这些条件是否相似。它们都有一个z,所以它们都一样。

Next, simplify them by adding the numerical part of the terms together. The $\begin{aligned} z \end{aligned}$ stays the same.
::接下来,通过将术语的数值部分加在一起来简化它们。 z 保持不变。

$\begin{aligned} 7 + 2 + 4 = 13 \end{aligned}$

Then, replace the $\begin{aligned} z \end{aligned}$ .
::然后,替换z。

So, $\begin{aligned} 7 z + 2 z + 4 z = 13 z \end{aligned}$
::7z + 2z + 4z = 13z

The answer is $\begin{aligned} 13 z \end{aligned}$ .
::答案是13兹

Example 4
::例4

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} 25 y - 13 y \end{aligned}$

First, look to see if these terms are alike. Both have a $\begin{aligned} y \end{aligned}$ , so they are alike.
::首先,看看这些条件是否相同。两个条件都有一个y,所以它们都一样。

Next, simplify them by adding the numerical part of the terms together. The $\begin{aligned} y \end{aligned}$ stays the same.
::接下来,通过将术语的数值部分加在一起来简化它们。y 保持不变。

$\begin{aligned} 25 - 13 = 12 \end{aligned}$

Then, replace the $\begin{aligned} y \end{aligned}$ .
::然后,替换Y。

$\begin{aligned} 12 y \end{aligned}$

The answer is $\begin{aligned} 12 y \end{aligned}$ .
::答案是12岁

Example 5
::例5

Simplify the following expression.
::简化以下表达式。

$\begin{aligned} 7 x + 2 x + 4 a \end{aligned}$

First, combine the like terms.
::首先,将类似条件结合起来。

$\begin{aligned} 7 x + 2 x \\ 4 a \end{aligned}$

Next, simplify the like terms by adding the numerical coefficients.
::接下来,简化类似术语,添加数字系数。

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

Then, replace the $\begin{aligned} x \end{aligned}$ .
::然后,替换 x 。

$\begin{aligned} 9 x \\ 4 a \end{aligned}$

The answer is $\begin{aligned} 9 x + 4 a \end{aligned}$ .
::答案是 9 x + 4 a 。

Review
::回顾

Simplify the following expressions by combining like terms. If the expression is already in simplest form, write “already in simplest form.”
::将类似术语合并来简化以下表达式。如果表达式已经以最简单的形式出现, 请写“ 已经以最简单的形式出现 ” 。

$\begin{aligned} 4 x + 6 x \end{aligned}$
::4 x + 6 x 4 x + 6 x

$\begin{aligned} 8 y + 5 y \end{aligned}$
::8 y + 5 y + 5 y

$\begin{aligned} 9 z + 2 z \end{aligned}$
::9 z + 2 z 9 z + 2 z

$\begin{aligned} 8 x + 2 y \end{aligned}$
::8 x + 2 yy 8 x + 2 y

$\begin{aligned} 7 y + 3 y + 2 x \end{aligned}$
::7y + 3y + 2x

$\begin{aligned} 9 x - x \end{aligned}$
::9 x - xx 9 x - x

$\begin{aligned} 12 y - 3 y \end{aligned}$
::12 y - 3 y 12 y - 3 y

$\begin{aligned} 22 x - 2 y \end{aligned}$
::22x-2yy 22x-2y

$\begin{aligned} 78 x - 10 x \end{aligned}$
::78x-10x

$\begin{aligned} 22 y - 4 y \end{aligned}$
::22 y - 4 y

$\begin{aligned} 16 x - 5 x + 1 x - 12 y + 2 y \end{aligned}$
::16 x - 5 x + 1 x - 12 y + 2 y

$\begin{aligned} 26 x - 15 x + 12 x - 14 y + 2 y \end{aligned}$
::26 x - 15 x + 12 x - 14 y + 2 y

$\begin{aligned} 36 x - 5 x + 11 x - 1 x + 2 y \end{aligned}$
::36 x - 5 x + 11 x - 1 x + 2 y

$\begin{aligned} 26 x - 25 x + 12 x - 13 y + 2 y \end{aligned}$
::26 x - 25 x + 12 x - 13 y + 2 y

$\begin{aligned} 29 x - 25 x + 18 x - 12 x + 12 y + 3 y \end{aligned}$
::29 x - 25 x + 18 x - 12 x + 12 + 12 y + 3 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

Simplifying Expressions Involving Sums and Differences ::涉及总和和差异的简化表达式

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 Expressions Involving Sums and Differences
::涉及总和和差异的简化表达式

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