Course: 12.3 通过类似术语的组合简化多义

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将类似术语的词串合并, 简化多义
Jessie is stuck on her math homework. She is stuck on the problem $\begin{align*}5x-3y-9x+7y\end{align*}$ . The directions are asking her to simplify, but she isn’t sure how to do that. Do you know?
::Jessie被困在数学功课上。她被困在5x-3y-9x+7的问题上。方向要求她简化,但她不知道如何简化。你知道吗?

In this concept, you will learn to simplify by combining like terms .
::在这个概念中,你会学会简化,把类似术语结合起来。

Combining Like Terms
::将类似术语合并

A polynomial is an algebraic expression that shows the sum of monomials. Since the prefix 'mono' means 'one', a monomial is a single piece or term . The prefix 'poly' means 'many'. So the word polynomial refers to multiple terms in an expression . The relationships between the terms may be sums or differences.
::多式表示法是一个代数表达式,它显示单项的和。由于前缀“ mono” 是指“ one ” , 单项表示单项或术语。前缀“ poly” 表示“ many ” 。因此,多式表示法是指一个表达式中的多个术语。术语之间的关系可以是数字或差异。

Polynomial expressions include: $\begin{align*}x^2+5 \qquad 3x-8+4x^5 \qquad \text{-}7a^2+9b-4b^3+6\end{align*}$
::复数表达式包括: x2+53x-8+4x5-7a2+9b-4b3+6

You can simplify polynomials by combining like terms. In mathematics, you are able to combine like terms but you cannot combine unlike terms.
::您可以通过将类似术语合并来简化多语种。在数学中,你可以将类似术语合并,但不能将不同术语合并。

Terms are considered like terms if they have exactly the same variables with exactly the same exponents.
::用语如具有完全相同的变量和完全相同的引言,则视为用语。

A term can also be a single number like 7 or -5. These are called constants .
::一个术语也可以是一个数字,如 7 或 - 5。这些称为常数。

Any term with a variable has a numerical factor called the coefficient . The coefficient of $\begin{align*}4x\end{align*}$ is 4. The coefficient of $\begin{align*}\text{-}7a^2\end{align*}$ is -7. The coefficient of $\begin{align*}y\end{align*}$ is 1 (because its numerical factor is an unwritten number 1. You could write “ $\begin{align*}1y\end{align*}$ ” to show that the coefficient of $\begin{align*}y\end{align*}$ is 1 but it is not necessary because any number multiplied by 1 is unchanged ).
::具有变量的任何术语都有一个数字系数。 4x系数为4。系数为 -7a2。系数为 -7。 y系数为 1( 因为它的数值系数是一个不成文的编号 1 ) 。您可以写“ 1y” 来显示y系数为 1, 但不必要, 因为任何数字乘以 1 不变。

Here are some examples of like and unlike terms:
::以下是一些类似和不同术语的例子:

$\begin{align*}7n\end{align*}$ and $\begin{align*}5n\end{align*}$ are like terms because they both have the variable $\begin{align*}n\end{align*}$ with an exponent of 1.
::7n和5n是相同的条件因为他们都有变量n 和1的指数

$\begin{align*}4n^2\end{align*}$ and $\begin{align*}\text{-}3n\end{align*}$ are not like terms because, although they both have the variable $\begin{align*}n\end{align*}$ , they do not have the same exponent.
::4n2 和 -3n 与术语不同,因为尽管它们都有变量 n,但它们没有相同的指数。

$\begin{align*}5x^3\end{align*}$ and $\begin{align*}8y^3\end{align*}$ are not like terms because, although they both have the same exponent, they do not have the same variable.
::5x3和8Y3与术语不同,因为尽管它们都有相同的指数,但它们没有相同的变量。

Like terms can be combined by adding their coefficients.
::类似术语可以通过增加其系数加以合并。

$\begin{aligned} \begin{array}{rcl} 7 n + 5 n & = & 12 n \\ 3 x^{3} + 5 x^{3} & = & 8 x^{3} \\ - 2 t^{4} - 10 t^{4} & = & - 12 t^{4} \\ 2 n^{2} - 3 n + 5 n^{2} + 11 n & = & 7 n^{2} + 8 n \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 7n + 5n &=& 12n \\ 3x^3 + 5x^3 &=& 8x^3 \\ -2t^4 - 10t^4 &=& -12t^4 \\ 2n^2 - 3n + 5n^2 + 11n &=& 7n^2 + 8n \end{array}\end{align*}$
::7n+5n=12n3x3+5x3=8x3_2t4-10t412t42n2-3n+5n2+11n=7n2+8n

Notice that the exponent does not change when you combine like terms. If you think of $\begin{align*}7n\end{align*}$ as simply a shorter way of writing $\begin{align*}n + n + n + n + n + n + n\end{align*}$ and $\begin{align*}5n\end{align*}$ as a shorter way of writing $\begin{align*}n + n + n + n + n,\end{align*}$ then combining those like terms would result in $\begin{align*}(n+n+n+n+n+n+n)+(n+n+n+n+n),\end{align*}$ which is the same as $\begin{align*}12n. \text{ So }7n+5n=12n.\end{align*}$
::请注意, 当您将类似条件合并时, 前名不会改变。如果您将 7n 简单地看作是写 n+n+n+n+n+n+n+n+n和 5n 的较短方式来写 n+n+n+n+n+n, 那么将这些类似条件合并起来将导致 (n+n+n+n+n+n+n+n)+(n+n+n+n+n+n+n), 这与 12n 相同。 So 7n+5n=12n 。

Examples
::实例

Example 1
::例1

Earlier, you were asked about helping Jessie with her simplification problem.
::之前有人问过你帮杰西处理简化问题

Here is the problem that Jessie is stuck on:
::这是杰西被困在下面的问题:

$\begin{aligned} 5 x - 3 y - 9 x + 7 y \end{aligned}$
$\begin{align*}5x - 3y - 9x + 7y\end{align*}$
::5-3y-9x+7y

First, consider the like terms and combine them:
::首先,考虑一下类似的术语,然后将它们结合起来:

$\begin{aligned} \begin{array}{rcl} 5 x - 9 x & = & - 4 x \\ - 3 y + 7 y & = & 4 y \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 5x - 9x &=& \text{-}4x \\ \text{-}3y + 7y &=& 4y \end{array}\end{align*}$
::5-9x=-4x-3y+7y=4y

Then write the terms in a single expression again :
::然后用一个词来写:

$\begin{aligned} - 4 x + 4 y \end{aligned}$
$\begin{align*}\text{-}4x + 4y\end{align*}$
::-4x+4y

The answer is $\begin{align*}\text{-}4x + 4y\end{align*}$ .
::答案是 - 4x+4y 。

Example 2
::例2

Simplify by combining like terms:
::将类似术语合并简化:

$\begin{aligned} 15 x - 12 x + 3 y - 8 x + 7 y - 1 + 5 \end{aligned}$
$\begin{align*}15x-12x+3y-8x+7y-1+5\end{align*}$
::15-12x+3y-8x+7y-1+5

First, let’s look at the like terms and combine them.
::首先,让我们看一看类似的术语,然后把它们结合起来。

$\begin{aligned} \begin{array}{rcl} 15 x - 12 x - 8 x & = & - 5 x \\ 3 y + 7 y & = & 10 y \\ - 1 + 5 & = & 4 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 15x - 12x - 8x &=& \text{-}5x \\ 3y + 7y &=& 10y \\ \text{-}1 + 5 &=& 4 \end{array}\end{align*}$
::15-12x-8x=-5x3y+7y=10y-1+5=4

Then rewrite the combined terms in a single expression:
::然后用一个单一的表达式重写合并术语 :

$\begin{aligned} - 5 x + 10 y + 4 \end{aligned}$
$\begin{align*}\text{-}5x+10y+4\end{align*}$
::-5x+10y+4

The answer is $\begin{align*}\text{-}5x+10y+4\end{align*}$ .
::答案是 - 5x+10y+4。

Example 3
::例3

Simplify by combining like terms.
::将类似术语合并简化。

$\begin{aligned} 2 x - 8 y - 4 x + 7 y + 9 \end{aligned}$
$\begin{align*}2x- 8y - 4x + 7y + 9\end{align*}$
::2 - 8y- 4x+7y+9

First, let’s look at the like terms and combine them where possible.
::首先,让我们看一看相似的术语,并尽可能将它们结合起来。

$\begin{aligned} \begin{array}{rcl} 2 x - 4 x & = & - 2 x \\ - 8 y + 7 y & = & - y \\ 9 & = & 9 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 2x-4x &=& \text{-}2x \\ \text{-}8y + 7y &=& \text{-}y \\ 9 &=& 9 \end{array}\end{align*}$
::2x-4x=-2x-8y+7y=-y9=9

Then rewrite the terms in a single expression:
::然后用一个单一的表达式重写术语 :

$\begin{aligned} - 2 x - y + 9 \end{aligned}$
$\begin{align*}\text{-}2x - y + 9\end{align*}$
::-2x-y+9

The answer is $\begin{align*}\text{-}2x - y + 9\end{align*}$ .
::答案是 - 2x-y+9。

Example 4
::例4

Simplify by combining like terms.
::将类似术语合并简化。

$\begin{aligned} 5 a + 3 b - 8 b + a - 7 \end{aligned}$
$\begin{align*}5a + 3b - 8b + a -7\end{align*}$
::5a+3b-8b+7

First, let’s look at the like terms and combine them.
::首先,让我们看一看类似的术语,然后把它们结合起来。

$\begin{aligned} \begin{array}{rcl} 5 a + a & = & 6 a \\ 3 b - 8 b & = & - 5 b \\ - 7 & = & - 7 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 5a + a &=& 6a \\ 3b - 8b &=& \text{-}5b \\ \text{-}7 &=& \text{-}7 \end{array}\end{align*}$
::5a+a=6a3b-8b=-5b-7=-7

Then rewrite the terms in a single expression:
::然后用一个单一的表达式重写术语 :

$\begin{aligned} 6 a - 5 b - 7 \end{aligned}$
$\begin{align*}6a-5b-7\end{align*}$
::6a-5b-7

The answer is $\begin{align*}6a-5b-7\end{align*}$ .
::答案是6a-5b-7。

Example 5
::例5

Simplify by combining like terms.
::将类似术语合并简化。

$\begin{aligned} 5 a - 7 b = 8 b - 2 a + 8 a - 9 + 8 \end{aligned}$
$\begin{align*}5a - 7b = 8b - 2a + 8a - 9 + 8\end{align*}$
::5a-7b=8b-2a+8a-9+8

First, look at the like terms and combine them.
::首先,看看相似的术语并结合它们。

$\begin{aligned} \begin{array}{rcl} 5 a - 2 a + 8 a & = & 11 a \\ - 7 b + 8 b & = & b \\ - 9 + 8 & = & - 1 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 5a - 2a + 8a &=& 11a \\ \text{-}7b + 8b &=& b \\ \text{-}9 + 8 &=& \text{-}1 \end{array}\end{align*}$
::5a-2a+8a=11a-7b+8b=b-9+8=-1

Then rewrite the combined terms in a single expression:
::然后用一个单一的表达式重写合并术语 :

$\begin{aligned} 11 a + b - 1 \end{aligned}$
$\begin{align*}11a + b - 1\end{align*}$
::11a+b-1

The answer is $\begin{align*}11a + b - 1\end{align*}$ .
::答案是11a+b-1。

Review
::回顾

Simplify the following polynomials by combining like terms.
::将类似术语合并,以简化以下多义词句。

$\begin{align*}6x + 7 - 18x + 4\end{align*}$
::6x+7-18x+4

$\begin{align*}5x - 7x + 5x + 4 -9\end{align*}$
::5-7x+5x+4-9

$\begin{align*}3x + 8y - 5x + 3y\end{align*}$
::3x+8y-5x+3y

$\begin{align*}17x^2 - 7x^2 - 5x + 3x + 14\end{align*}$
::17x2 - 7x2 - 5x+3x+14

$\begin{align*}3xy - 9xy - 5x + 4x - 7 + 3\end{align*}$
::3xy-9xy-5x+4x-7+3 3xy-9xy-5xx+4x-7+3

$\begin{align*}9x + 7y - 15x + 4x - 9y\end{align*}$
::9x+7y-15x+4x-9y

$\begin{align*}3x + 7 - 5x - 8y + 4x - 2y + 7\end{align*}$
::3x+7-5x-8y+4x-2y+7

$\begin{align*}3xy - xy - 15x + 4 -11\end{align*}$
::3xy-xy-15x+4-11

$\begin{align*}\text{-}8x + 3x + 7y - 5x + 4y - 2\end{align*}$
::-8x+3x+7y-5x+4y-2

$\begin{align*}3x^2 + 6x - 3y + 2x - 7\end{align*}$
::3x2+6x-3y+2x-7

$\begin{align*}14xy - 18xy + 7y + 8x - 2x + 9\end{align*}$
::14xy-18xy+7y+8x-2x+9

$\begin{align*}3x + 7 - 5x + 4y - 18y\end{align*}$
::3x+7-5x+4y-18y

$\begin{align*}6y^2 - 4y^3 + y^2 - 8\end{align*}$
::6y2-4y3+y2-8

$\begin{align*}\text{-}5q + q^2 + 7 - q -7\end{align*}$
::-5q+q2+7-q-7

$\begin{align*}n^2m - 3n^2m + 5n^2m^2 + 11n\end{align*}$
::n2m-3n2m+5n2m2+11n

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

将类似术语的词串合并, 简化多义

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

Example 4 ::例4

Example 5 ::例5

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

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)