Course: 1.5 简化代数表达式

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简化代数表达式
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

S ometimes variables and numbers can be repeated within an expression . If the same variable is in an expression more than once, the occurrences can be combined by addition or subtraction . This process is called combining like terms.
::有时变量和数字可以在表达式中重复出现。如果同一个变量在一个表达式中出现不止一次,则可以通过增加或减法将发生的情况合并。此过程被称为类似术语的组合。

Let's simplify each of the following expressions.
::让我们简化以下表达式的每一个表达式。

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

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 parenthesis around the like terms.
::重组表达式以将 x 和数字组合在一起。您可以将类似术语放在每个词旁边, 或者将类似术语括号放在类似术语周围。

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

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

Notice that the Greatest Common Factor (GCF) for $\begin{aligned} 2 x \end{aligned}$ and 8 is 2. Therefore , you can use the to pull out the GCF to get $\begin{aligned} 2 (x - 4) \end{aligned}$ .
::注意2x和8的最大共同系数(GCF)是2。因此,你可以利用这一系数来拉出全球合作框架,以获得2(x-4)。

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

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.
::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)

There is only one number term , called the constant , so we leave it at the end. Also, in general, list the variables in alphabetical order .
::只有一个数字术语,称为常数,所以我们将其留在结尾处。此外,一般地,按字母顺序列出变量。

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。

Here we have one variable, but there are different powers (exponents). Like terms must have the same exponent in order to combine them.
::这里我们有一个变量, 但有不同的权力( explents ) 。类似地, 术语必须具有相同的引号才能合并它们。

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

::w2+9-4w2+3w2+4w2+3w2+3w4-7w4-1113w4+(w2-4w2)-7w2+(9-11)3w4-3w2-7w-2

When writing an expression with different powers, list the powers from greatest to least, like above.
::在以不同权力写出一种表达方式时,请列出从最大到最小的权力,如上文。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the total number of apples, oranges, and limes Corey and Katelyn have all together.
::早些时候,你被要求找到苹果、橙子和石灰科里和凯特林的总数

Let's rewrite Corey's bowl of fruit as $\begin{aligned} 5 a + 4 o + 3 l \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ represents apples, $\begin{aligned} o \end{aligned}$ represents oranges, and $\begin{aligned} l \end{aligned}$ represents limes. Then Katelyn's bowl of fruit can be represented using the same format as $\begin{aligned} 2 a + 5 l + o \end{aligned}$ . Combining like terms, we have:
::让我们重写科里(Corey)的水果碗为 5a+4o+3l, 其中代表苹果, O 代表橙子, I 代表石灰。然后凯特林(Katelyn)的水果碗可以用 2a+5l+o 相同的格式来表示。结合类似条件, 我们有:

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

:5a+4o+3l)+(2a+5l+5a+2a)+(4o+o)+(3l+5l)7a+5o+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 2
::例2

Simplify the following expression: $\begin{aligned} 6 s - 7 t + 12 t - 10 s \end{aligned}$ .
::简化以下表达式: 6s-7t+12t- 10s。

Combine the $\begin{aligned} s \end{aligned}$ ’s and the $\begin{aligned} t \end{aligned}$ ’s.
::将两者结合起来。

$\begin{aligned} 6 s - 7 t + 12 t - 10 s \\ (6 s - 10 s) + (- 7 t + 12 t) \\ - 4 s + 5 t \end{aligned}$

::6-7t+12t-10s(6s-10s)+(-7t+12t)-4s+5t

Notice in that we did not write $\begin{aligned} (6 s - 10 s) - (7 t + 12 t) \end{aligned}$ in the second step. This would lead us to an incorrect answer. Whenever grouping together like terms , if one is negative (or being subtracted), always change the operator to addition and make the subtracted number negative. In other words, when applying the commutative property to reorganize an expression, keep the sign to the left of a term with the term, then you can remove the extraneous +/- signs after you have everything sorted, like this:
::请注意, 我们在第二步没有写入( 6s- 10s)-( 7t+12t)-( 6s- 10s)-( 7t+12t) 。这将引导我们找到一个错误的答案。当像条件一样组合在一起时, 如果一个条件为负( 或被减去) , 总是会将操作者更改为加号, 减数为负数。换句话说, 当应用通货属性重组表达式时, 将标记保留在句子左端时, 然后您就可以在全部排序完毕后删除不相干 +/ 符号, 比如 :

$\begin{aligned} 6 s - 7 t + 12 t - 10 s \\ (6 s) + (- 7 t) + (+ 12 t) + (- 10 s) \\ (6 s) + (- 10 s) + (+ 12 t) + (- 7 t) \\ 6 s - 10 s + 12 t - 7 t \\ - 4 s + 5 t \end{aligned}$

::6-7t+12t-10s(6)+(7)+(12)+(12)+(10)+(10)+(10)+(10)+(12)+(12)+(7)+(7)+(7)6s-10s+12t-4s+5t

Example 3
::例3

Simplify the following expression: $\begin{aligned} 7 y^{2} - 9 x^{2} + y^{2} - 14 x + 3 x^{2} - 4 \end{aligned}$ .
::简化以下表达式: 7y2- 9x2+y2-14x+3x2-4。

Group together the like terms.
::将类似术语组合在一起。

$\begin{aligned} 7 y^{2} - 9 x^{2} + y^{2} - 14 x + 3 x^{2} - 4 \\ (- 9 x^{2} + 3 x^{2}) + (7 y^{2} + y^{2}) - 14 x - 4 \\ - 6 x^{2} + 8 y^{2} - 14 x - 4 \end{aligned}$

::7y2-9x2+y2-14x+3x2-4(9x2+3x2)+(7y2+y2)-14x-4-6x2+8y2-14x-4

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 ” 。

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

$\begin{aligned} 6 - 11 c + 5 c - 18 \end{aligned}$
::6-11c+5c-18

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

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

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

$\begin{aligned} 7 p - p^{2} + 9 p + q^{2} - 16 - 5 q^{2} + 6 \end{aligned}$
::7-p2+9p+q2-16-5q2+6

$\begin{aligned} 20 x - 6 - 13 x + 19 \end{aligned}$
::20-6-13x+19 20-6-6-13x+19

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

Find the GCF of the following expressions and use the Distributive Property to simplify each one.
::查找以下表述方式的全球合作框架,并利用分配财产来简化每个表述方式。

$\begin{aligned} 6 a - 18 \end{aligned}$
::6a-18

$\begin{aligned} 9 x^{2} - 15 \end{aligned}$
::9x2-15

$\begin{aligned} 14 d + 7 \end{aligned}$
::14d+7 14d+7

$\begin{aligned} 3 x - 24 y + 21 \end{aligned}$
::3x-24y+21

Challenge We can also use the Distributive Property and GCF to pull out common variables from an expression. Find the GCF and use the Distributive Property to simplify the following expressions.
::我们还可以利用分配财产和绿色气候基金来从一种表达方式中提取共同变量。找到绿色气候基金并利用分配财产来简化以下表达方式。

$\begin{aligned} 2 b^{2} - 5 b \end{aligned}$
::2b2-5b

$\begin{aligned} m^{3} - 6 m^{2} + 11 m \end{aligned}$
::m3-6m2+1 100米

$\begin{aligned} 4 y^{4} - 12 y^{3} - 8 y^{2} \end{aligned}$
::4y4 - 12y3 - 8y2

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。

Section outline

General

简化代数表达式

Combining Like Terms ::将类似术语合并

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Combining Like Terms
::将类似术语合并

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)