Section: 简化代数表达式 | 1.14 简化代数表达式

Section outline

Corey has a bowl of fruit that consists of 5 apples, 4 oranges , and 3 limes . Katelyn went to the farmer's market and picked up 2 apples, 5 limes , and an orange. How many apples, oranges , and limes do Corey and Katelyn have combined?
::科里有一碗水果,由5个苹果、4个橙子和3个石灰组成。凯特林去了农贸市场,拿了2个苹果、5个石灰和1个橙子。科里和凯特林加了多少苹果、橙子和石灰?

Combining like terms is much like grouping together different fruits, like apples and oranges.
::结合类似术语就像是将不同的水果组合在一起,比如苹果和橙子。

Combining Like Terms
::将类似术语合并

You might have noticed from the previous concept, that sometimes variables and numbers can be repeated within an expression . If the same variable appears in an expression more than once, the terms with that variable possibly can be combined by addition or subtraction . This process is called combining like terms .
::您可能已经从上一个概念中注意到, 有时变量和数字可以在表达式中重复。如果同一个变量出现在一个表达式中不止一次, 与该变量的术语可能可以通过添加或减法合并。这个过程被称为类似术语的组合。

Before we discuss combining like terms, let's first look at what like terms are.
::在我们讨论将类似术语合并之前, 让我们先看看术语是什么样的。

Like Terms and How to Combine Them
::类似术语和如何将两者结合起来

Like terms are terms that have the same variables and each variable is raised to the same power.
::类似术语是具有相同变数的术语,每个变数被提升到相同的变数。

To combine like terms, add or subtract the coefficients and keep the variable part of the terms .
::将类似术语合并,增减系数,保留术语的可变部分。

Example 1
::例1

Are the following pairs of terms like terms?
::以下的两条术语是否类似术语?

a. $\begin{aligned} 4 x y^{2} z^{6} \end{aligned}$ and $\begin{aligned} - 3 x y^{2} z^{6} \end{aligned}$
::a. 4xy2z6和-3xy2z6

b. $\begin{aligned} 4 x y^{2} z^{6} \end{aligned}$ and $\begin{aligned} 7 x y^{2} \end{aligned}$
::b. 4xy2z6和7xy2

c. $\begin{aligned} 4 x y^{2} z^{6} \end{aligned}$ and $\begin{aligned} - 2 x^{2} y z^{6} \end{aligned}$
::c. 4xy2z6和-2x2yz6

Solution:
::解决方案 :

a. To determine if two terms are like terms, we need to focus on their variable parts. Each term has an $\begin{aligned} x \end{aligned}$ , a $\begin{aligned} y \end{aligned}$ , and a $\begin{aligned} z \end{aligned}$ . In each, $\begin{aligned} x \end{aligned}$ is raised to the first power, $\begin{aligned} y \end{aligned}$ is raised to the second power, and $\begin{aligned} z \end{aligned}$ is raised to the sixth power. Since these terms have the same variables and each is raised to the same power, they are like terms and we can combine them.
::a. 为了确定两个术语是否类似术语,我们需要关注其变量部分。每个术语有一个 x, a y 和 z。每个术语有一个 x, a y 和 z。在每一个术语中, x 被提升到第一个电源, y 被提升到第二个电源, z 被提升到第六个电源。由于这两个术语具有相同的变量,而每个变量被提升到同一个电源,它们就像术语,我们可以将它们结合起来。

b. While both of these terms have an $\begin{aligned} x \end{aligned}$ and a $\begin{aligned} y \end{aligned}$ , $\begin{aligned} 7 x y^{2} \end{aligned}$ does not have $\begin{aligned} z \end{aligned}$ as a variable. Therefore , they are not like terms and we cannot combine them.
::b. 虽然这两个术语都有 x 和 y, 7xy2 没有z作为变量。因此,它们与术语不同,我们不能将它们合并。

c. These two terms have the same variables, but the $\begin{aligned} x \end{aligned}$ is raised to the first power in the first term and raised to the second power in the second term. The powers for $\begin{aligned} y \end{aligned}$ are not the same as well. Therefore, these are not like terms and cannot be combined.
::c. 这两个词具有相同的变量,但x在第一个任期内被提升到第一个权力,在第二个任期内被提升到第二个权力。y的权力也不相同,因此,它们与术语不同,不能合并。

Example 2
::例2

Simplify $\begin{aligned} 5 x - 12 - 3 x + 4 \end{aligned}$ .
::简化 5x- 12- 3x+ 4。

Solution: To simplify here, reorganize the expression to group together the $\begin{aligned} x \end{aligned}$ ’s and the numbers. You can either place the like terms next to each together or place " data-term="Parentheses" role="term" tabindex="0"> parentheses around the like terms.
::解决方案 : 要在此简化, 请将表达式重新组合为 x 和数字组。您可以将类似条件放在彼此旁边, 或者将类似条件括号放在类似条件周围。

$\begin{aligned} 5 x - 12 - 3 x + 4 \\ 5 x - 3 x - 12 + 4 or (5 x - 3 x) + (- 12 + 4) \\ 2 x - 8 \end{aligned}$

::5-12-3x+45x-3x-12+4or(5x-3x)+(-12+4)2x-8

Example 3
::例3

Simplify $\begin{aligned} 6 a - 5 b + 2 a - 10 b + 7 \end{aligned}$ .
::简化 6a- 5b+2a- 10b+7。

Solution: Here there are two different variables, $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ . Even though they are both variables, they are different variables and cannot be combined. Group together the like terms.
::解决办法:这里有两个不同的变量,a和b。尽管这两个变量都是变量,但它们是不同的变量,不能合并。将类似术语组合在一起。

$\begin{aligned} 6 a - 5 b + 2 a - 10 b + 7 \\ = (6 a + 2 a) + (- 5 b - 10 b) + 7 \\ = 8 a - 15 b + 7 \end{aligned}$

::6-5b+2a-10b+7=(6a+2a)+(-5b-10b)+7=8a-15b+7

The term that is just a number is called the constant term and we tend to place it last . Also, in general, list the variables in alphabetical order .
::仅是一个数字的术语被称为常数术语,我们倾向于将其置于最后。此外,一般地,用字母顺序列出变量。

Example 4
::例4

Simplify $\begin{aligned} w^{2} + 9 - 4 w^{2} + 3 w^{4} - 7 w - 11 \end{aligned}$ .
::简化 w2+9-4w2+3w4-7w-11。

Solution: Here we have one variable, but there are different powers. We need to identify the like terms to combine them.
::解决办法:我们有一个变量,但权力不同。我们需要找出相似的术语来将它们结合起来。

$\begin{aligned} w^{2} + 9 - 4 w^{2} + 3 w^{4} - 7 w - 11 \\ = 3 w^{4} + (w^{2} - 4 w^{2}) - 7 w + (9 - 11) \\ = 3 w^{4} - 3 w^{2} - 7 w - 2 \end{aligned}$

::w2+9-4w2+3w2+4w2+3w2+3w4-7w-7w-11=3w4+(w2-4w2)-7w+(9-11)=3w4-3w2-7w-2)

When writing an expression with different powers, we tend to list the powers from greatest to least, as we did above.
::在以不同权力写出表达方式时,我们倾向于像上文那样,从最大到最小地列出权力。

by Mathispower4u demonstrates how to simplify algebraic expressions.
::用 Mathispower4u 演示如何简化代数表达式。

Example 5
::例5

Corey has a bowl of fruit that consists of 5 apples, 4 oranges , and 3 limes . Katelyn went to the farmer's market and picked up 2 apples, 5 limes , and 1 orange . How many apples, oranges , and limes do Corey and Katelyn have combined?
::科里有一碗水果,由5个苹果、4个橙子和3个石灰组成。凯特林去了农贸市场,拿了2个苹果、5个石灰和1个橙子。科里和凯特林加了多少苹果、橙子和石灰?

Solution: Let's rewrite Corey's bowl of fruit as $\begin{aligned} 5 a + 4 r + 3 l \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ represents apples, $\begin{aligned} r \end{aligned}$ represents oranges , and $\begin{aligned} l \end{aligned}$ represents limes . Then Katelyn's bowl of fruit can be represented as $\begin{aligned} 2 a + 5 l + r \end{aligned}$ . Combining like terms, we have:
::解答:让我们重写科里(Corey)的水果碗5a+4r+3l, 代表苹果, R代表橙子, I代表石灰。然后凯特林(Katelyn)的水果碗可以代表2a+5l+r。结合类似条件, 我们有:

$\begin{aligned} (5 a + 4 r + 3 l) + (2 a + 5 l + r) \\ = (5 a + 2 a) + (4 r + r) + (3 l + 5 l) \\ = 7 a + 5 r + 8 l \end{aligned}$

:5a+4r+3l)+(2a+5l+r)=(5a+2a)+(4r+r)+(3l+5l)=7a+5r+8l

Together they have 7 apples, 5 oranges , and 8 limes .
::他们共有7个苹果,5个橙子,8个石灰。

Example 6
::例6

Jane needs to buy a notebook, a folder, and a USB drive for each of the five classes she is taking this semester. She also needs to buy two additional notebooks for her extra-curricular activities. If we let n = the cost of a notebook, f = the cost of a folder, and u = the cost of a USB drive, we could use the following expression to represent the amount she will spend on these supplies: $\begin{aligned} 5 (n + f + u) + 2 n \end{aligned}$ . Simplify this expression.
::Jane需要购买笔记本、文件夹和USB驱动器, 供她本学期学习的五个班级中的每一个班级使用。她还需要为她的课外活动另外购买两本笔记本。如果我们让 n = 笔记本的费用, f = 文件夹的费用, u = USB驱动器的费用, 我们可以使用以下表达式来表示她将在这些用品上花费的金额: 5 (n+f+u)+2n。简化这个表达式。

Solution: We need to first distribute the 5. We will multiply 5 by n, f, and u. Then we will combine the like terms.
::解决方案:我们首先需要分配5,我们将乘以5乘以n,f,u。然后,我们将合并同样的术语。

$\begin{aligned} 5 (n + f + u) + 2 n \\ = 5 n + 5 f + 5 u + 2 n \\ = (5 n + 2 n) + 5 f + 5 u \\ = 7 n + 5 f + 5 u \end{aligned}$

::5(n+f+u)+2n=5n+5f+5u+2n=(5n+2n)+5f+5u=7n+5f+5u

Summary
::摘要

Like terms are terms that have the same variables and each variable is raised to the same power.
::类似术语是具有相同变数的术语,每个变数被提升到相同的变数。

::类似术语是具有相同变数的术语,每个变数被提升到相同的变数。

To combine like terms, add or subtract the coefficients and keep the variable part of the terms.
::将类似术语合并,增减系数,保留术语的可变部分。

::将类似术语合并,增减系数,保留术语的可变部分。

To simplify algebraic expressions, perform any operations in parentheses, distribute where possible, and then add and subtract like terms from left to right.
::为了简化代数表达式,在括号中进行任何操作,尽可能进行分配,然后从左向右增减类似术语。

Review
::回顾

Simplify the following expressions as much as possible. If the expression cannot be simplified, write “cannot be simplified.”
::尽可能简化以下表达式。如果该表达式不能简化, 请写“ 不能简化 ” 。

1. $\begin{aligned} 5 b - 15 b + 8 d + 7 d \end{aligned}$
::1. 5b-15b+8d+7d

2. $\begin{aligned} 3 g^{2} - 7 g^{2} + 9 + 12 \end{aligned}$
::2. 3g2-7g2+9+12

3. $\begin{aligned} 8 u^{2} + 5 u - 3 u^{2} - 9 u + 14 \end{aligned}$
::3. 8u2+5u-3u2-9u+14

4. $\begin{aligned} 2 a - 5 f \end{aligned}$
::4.a-5f

5. $\begin{aligned} 7 p - p^{2} + 9 p + q^{2} - 16 - 5 q^{2} + 6 \end{aligned}$
::5. 7p-2+9p+q2-16-5q2+6

6. $\begin{aligned} 8 n - 2 - 5 n^{2} + 9 n + 14 \end{aligned}$
::6. 8n-2-5n2+9n+14

7. $\begin{aligned} \frac{4 r^{2} s^{5}}{r s^{3}} \end{aligned}$
::7. 4r2s5rs3

8. $\begin{aligned} \frac{7 c^{3}}{11 d} \end{aligned}$
::8. 7c311d

9. $\begin{aligned} \frac{6 a - 18}{3} \end{aligned}$
::9. 6a-183

10. $\begin{aligned} \frac{2 b^{2} - 5 b}{b} \end{aligned}$
::10. 2b2-5bbb

11. $\begin{aligned} 3 (4 x - 5) + 2 x \end{aligned}$
::11. 3(3,4x-5)+2x

12. $\begin{aligned} 7 (2 x + 3) - 6 x \end{aligned}$
::12. 7(2x+3)-6x

13. $\begin{aligned} - 4 (x - 5) + 2 x + 5 (3 - 2 x) \end{aligned}$
::13.-4(x-5)+2x+5(3-2x)

14. $\begin{aligned} 2 (5 + 8 x) - 6 x + 9 (3 x + 1) \end{aligned}$
::14. 2(5+8x)-6x+9(3x+1)

Explore More
::探索更多

1. Combine like terms and u se the rules of exponents to simplify the following expression:
::1. 将类似术语合并,使用推手规则简化以下表述:

$\begin{aligned} \frac{4 x^{2} y z^{3} + 5 x^{2} y z^{3} - 12 x^{2} y z^{3}}{x y^{2} z - 2 x y^{2} z} \end{aligned}$

::4x2yz3+5x2yz3-12x2yz3x2yz3x2z2z-2x2z

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try these interactives that reinforce the concepts explored in this section:
::尝试这些强化本节所探讨概念的交互作用 :

Section outline

Combining Like Terms ::将类似术语合并

Like Terms and How to Combine Them ::类似术语和如何将两者结合起来

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

Combining Like Terms
::将类似术语合并

Like Terms and How to Combine Them
::类似术语和如何将两者结合起来

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX