Course: 4.6简化包含整数加法的变量表达式

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简化包含整数加法的变量表达式
Ashley has just started a new business making and selling granola bars. So far she has been selling the granola bars at soccer games on Saturdays. Ashley sold 10 bars in her first week of business and 25 bars in her second week of business. If $\begin{aligned} b \end{aligned}$ represents the profit she makes from selling one bar, how could Ashley write and simplify a variable expression that represents her current profit after two weeks in business?
::阿什利刚刚开始一个新的商业生产,并出售了麦片棒。到目前为止,她一直在星期六的足球赛上出售麦片棒。阿什利在她第一周的生意中出售了10个酒吧,在第二周的生意中出售了25个酒吧。如果b 代表她从卖一个酒吧中获得的利润,那么阿什利怎么能写和简化一个代表她两周后目前利润的可变表达式呢?

In this concept, you will learn how to simplify variable expressions involving integer addition .
::在此概念中,您将学会如何简化包含整数添加的变量表达式。

Simplifying Variable Expressions Involving Integer Addition
::包含整数添加的简化变量表达式

An expression is a number phrase that contains numbers and operations .
::表达式是一个包含数字和操作的数句。

Here are some examples of expressions:
::以下是一些表达方式的例子:

$\begin{aligned} 2 x - 4 \end{aligned}$
::2 x - 4 个

$\begin{aligned} - 15 + 7 - 1 \end{aligned}$

$\begin{aligned} 3 x + 5 y - 2 \end{aligned}$
::3 x + 5 yy - 2 3 x + 5 Yy - 2

$\begin{aligned} 5 x + 2 x \end{aligned}$
::5 x + 2 x 5 x + 2 x

A variable is a symbol or letter (often $\begin{aligned} x \end{aligned}$ or $\begin{aligned} y \end{aligned}$ ) that is used to represent one or more numbers. A variable expression is an expression that includes variables.
::变量是指用于代表一个或多个数字的符号或字母(通常为 x 或 y )。变量表达式是指包含变量的表达式。

Here are some examples of variable expressions:
::以下是一些变量表达式的例子:

$\begin{aligned} 2 x - 4 \end{aligned}$
::2 x - 4 个

$\begin{aligned} 3 x + 5 y - 2 \end{aligned}$
::3 x + 5 yy - 2 3 x + 5 Yy - 2

$\begin{aligned} 5 x + 2 x \end{aligned}$
::5 x + 2 x 5 x + 2 x

In a variable expression, like terms are two terms that include the same variable. If a variable expression has like terms, it can be simplified by combining the like terms into one single term .
::在变量表达式中,类似术语是包含相同变量的两个术语。如果变量表达式有类似术语,可以通过将类似术语合并成一个单一术语来简化该术语。

Here is an example.
::举一个例子。

Simplify $\begin{aligned} 5 x + 2 x \end{aligned}$ .
::简化 5 x + 2 x 。

In this variable expression, $\begin{aligned} 5 x \end{aligned}$ and $\begin{aligned} 2 x \end{aligned}$ are like terms because they include the same variable, $\begin{aligned} x \end{aligned}$ .
::在此变量表达式中, 5 x 和 2 x 是相似的术语, 因为它们包含相同的变量 x 。

To simplify the expression, focus on the coefficients , which are the numbers in front of the $\begin{aligned} x \end{aligned}$ 's. In this case, your coefficients are 5 and 2. Add them up in order to simplify the variable expression.
::为简化表达式, 聚焦于系数, 即 x 前面的数字。在这种情况下, 您的系数是 5 和 2 。添加它们是为了简化变量表达式。

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

Think of it as if you had 5 $\begin{aligned} x \end{aligned}$ 's and 2 more $\begin{aligned} x \end{aligned}$ 's, you'd have 7 $\begin{aligned} x \end{aligned}$ 's total.
::想想看,如果你有5x和2x, 你会有7x的总数。

Here is another example.
::下面是另一个例子。

$\begin{aligned} 9 y + 8 x \end{aligned}$
::9 y + 8 x 9 y + 8 x

This variable expression cannot be combined or simplified. It does not have like terms. The $\begin{aligned} x \end{aligned}$ and the $\begin{aligned} y \end{aligned}$ are different, so you cannot do anything with this expression. It is in simplest form.
::此变量表达式不能合并或简化。它没有相似的术语。 x 和 y 不同, 因此您无法用此表达式做任何事情。它以最简单的形式出现。

You can use what you know about adding integers to help you to simplify variable expressions.
::您可以使用您对添加整数的了解来帮助您简化变量表达式。

Let's look at one more example.
::让我们再举一个例子。

Find the sum of $\begin{aligned} - 3 t + 9 t \end{aligned}$ .
::查找 - 3 t + 9 t 的总和。

Since $\begin{aligned} - 3 t \end{aligned}$ and $\begin{aligned} 9 t \end{aligned}$ both have the same variable, they are like terms. Use what you know about how to add integers to help you add the terms.
::因为 3 t 和 9 t 都有相同的变量, 它们就像术语。使用您知道的如何添加整数来帮助您添加术语。

Notice that the two like terms have different signs. So, your first step is to find the absolute values of both integers. Then subtract the term whose integer has the lesser absolute value from the other term.
::注意两个类似术语有不同的符号。所以, 您的第一步是找到两个整数的绝对值。然后减去一个整数与另一个整数的绝对值相比较小的术语。

$\begin{aligned} | - 3 | = 3 \end{aligned}$ and $\begin{aligned} | 9 | = 9 \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\begin{aligned} 9 - 3 = 6 \end{aligned}$

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original term that had the greater absolute value. Since 9 is greater than 3, and $\begin{aligned} 9 t \end{aligned}$ has a positive sign, give the answer a positive sign.
::您的下一步是决定最后答案上应该写什么符号。您的答案应该写上原术语的符号, 其绝对值更大。因为9 大于 3, 9 t 有积极的迹象, 请给答案一个积极的迹象。

The answer is $\begin{aligned} - 3 t + 9 t = 6 t \end{aligned}$ .
::答案是 - 3 t + 9 t = 6 t 。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Ashley and her granola bar business.
::早些时候,你得到一个问题关于阿什利和她的粮仓酒吧业务。

Her first week in business she sold 10 bars and her second week in business she sold 25 bars. She wants to come up with a variable expression to represent her total profit so far.
::她的第一周在生意上卖了10个酒吧,第二周在生意上卖了25个酒吧,她想用一个变数表达方式来表示她迄今的全部利润。

If $\begin{aligned} b \end{aligned}$ represents her profit from one bar, then in the first week her profit was $\begin{aligned} 10 b \end{aligned}$ and in the second week her profit was $\begin{aligned} 25 b \end{aligned}$ . To find her total profit, she wants to simplify
::如果b表示她从一家酒吧获得利润,那么第一周她的利润是10 b,第二周的利润是25 b。

$\begin{aligned} 10 b + 25 b \end{aligned}$
::10 b+25 b

To simplify this expression, start by looking at the signs of the terms. $\begin{aligned} 10 b \end{aligned}$ is positive and $\begin{aligned} 25 b \end{aligned}$ is also positive. Because they have the same sign, you will need to add their absolute values.
::要简化这个表达式,首先从术语的符号开始。 10 b 是正值, 25 b 也是正值。由于它们有相同的符号, 您需要添加它们的绝对值。

$\begin{aligned} | 10 | = 10 \end{aligned}$ and $\begin{aligned} | 25 | = 25 \end{aligned}$ .
::10 = 10 = 25 = 25 = 25 = 10 = 10 = 25 = 25 = 25 =

$\begin{aligned} 10 + 25 = 35 \end{aligned}$

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original terms. Since the original terms were positive, your final answer should be positive.
::您的下一步是决定您最后答案上应该写什么标志。您的答案应该写上原始条件的标志。由于最初条件是肯定的, 您的最后答案应该是肯定的。

The answer is $\begin{aligned} 10 b + 25 b = 35 b \end{aligned}$ .
::答案是10 b+25 b=35 b。

Ashley's profit so far is $\begin{aligned} 35 b \end{aligned}$ , where $\begin{aligned} b \end{aligned}$ represents the profit she makes from selling one granola bar.
::Ashley的利润目前为35美元b, B代表她出售一个麦片棒的利润。

Example 2
::例2

Simplify $\begin{aligned} 7 z + (- 3 z) \end{aligned}$ .
::简化 7 z + ( - 3 z) 。

First, notice that both terms have the same variable, $\begin{aligned} z \end{aligned}$ , so they are like terms.
::首先,请注意这两个词的变量是相同的,z,所以它们就像术语。

Next, look at the signs of the terms. $\begin{aligned} 7 z \end{aligned}$ is positive, but $\begin{aligned} - 3 z \end{aligned}$ is negative. Because they have different signs, you will need to subtract their absolute values.
::接下来,请看术语的符号。 7 z 是正值, 但 - 3 z 是负值。因为有不同的符号, 您需要减去它们的绝对值。

$\begin{aligned} | 7 | = 7 \end{aligned}$ and $\begin{aligned} | - 3 | = 3 \end{aligned}$ .
::7 = 7 = 7 = 3 = 3 = 3 = 7 = 7 = 3 = 3 = 3

$\begin{aligned} 7 - 3 = 4 \end{aligned}$

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original term that had the greater absolute value. Since 7 is greater than 3, and $\begin{aligned} 7 z \end{aligned}$ has a positive sign, give the answer a positive sign.
::您的下一步是决定最后答案上应该写什么符号。您的答案应该写上原术语的符号, 其绝对值更大。因为 7 大于 3 , 7 z 有积极信号, 请给答案一个积极信号。

The answer is $\begin{aligned} 7 z + (- 3 z) = 4 z \end{aligned}$ .
::答案是 7 z + ( - 3 z) = 4 z 。

Example 3
::例3

Simplify $\begin{aligned} - 8 x + (- 5 x) \end{aligned}$ .
::简化 - 8 x + ( - 5 x ) 。

First, notice that both terms have the same variable, $\begin{aligned} x \end{aligned}$ , so they are like terms.
::首先,请注意这两个词的变量是相同的, x, 所以它们是相似的术语。

Next, look at the signs of the terms. $\begin{aligned} - 8 x \end{aligned}$ is negative and $\begin{aligned} - 5 x \end{aligned}$ is also negative. Because they have the same sign, you will need to add their absolute values.
::接下来,请看术语的符号。 - 8 x 是负的, - 5 x 也是负的。由于它们有相同的符号, 您需要添加它们的绝对值。

$\begin{aligned} | - 8 | = 8 \end{aligned}$ and $\begin{aligned} | - 5 | = 5 \end{aligned}$ .
::=8 = 8 = 和 = 5 = 5 = 5 = = = = = = = = = = = = = = = = = = = = = = = = = = 5 = = = = = = = = = = = = = = = = = = = = = = = = 5 = = = = = = = = = = = = = = = = = 5 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

$\begin{aligned} 8 + 5 = 13 \end{aligned}$

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original terms. Since the original terms were negative, your final answer should be negative.
::您的下一步是决定您最后答案上应该写什么标志。您的答案应该写上原始条件的标志。由于原来的条件是否定的, 您的最后答案应该是否定的。

The answer is $\begin{aligned} - 8 x + (- 5 x) = - 13 x \end{aligned}$ .
::答案是 - 8 x + ( - 5 x ) = - 13 x 。

Example 4
::例4

Simplify $\begin{aligned} - 19 y + 5 y \end{aligned}$ .
::简化 - 19 y + 5 y 。

First, notice that both terms have the same variable, $\begin{aligned} y \end{aligned}$ , so they are like terms.
::首先,请注意这两个词的变量是相同的,y,所以它们就像术语。

Next, look at the signs of the terms. $\begin{aligned} - 19 y \end{aligned}$ is negative, but $\begin{aligned} 5 y \end{aligned}$ is positive. Because they have different signs, you will need to subtract their absolute values.
::接下来,请看术语的符号。 - 19 y 是负值, 但 5 y 是正值。由于它们有不同的符号, 您需要减去它们的绝对值。

$\begin{aligned} | - 19 | = 19 \end{aligned}$ and $\begin{aligned} | 5 | = 5 \end{aligned}$ .
::= 19 = 19 = 5 = 5 = 5 = 5 = = = 19 = 19 = = 5 = = 5 = = 5 = = = = = = = = = = = = = = = 5 = = = = 5 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 5 = = = = = 5 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 5 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

$\begin{aligned} 19 - 5 = 14 \end{aligned}$

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original term that had the greater absolute value. Since 19 is greater than 5, and $\begin{aligned} - 19 y \end{aligned}$ has a negative sign, give the answer a negative sign.
::您的下一步是决定最后答案上应该写什么符号。您的答案应该写上原术语的符号, 其绝对值更大。因为19岁比5岁大, — 19 y 有否定的符号, 请给答案一个否定的符号。

The answer is $\begin{aligned} - 19 y + 5 y = - 14 y \end{aligned}$ .
::答案是 - 19 y + 5 y = 14 y 。

Example 5
::例5

Simplify $\begin{aligned} - 6 y + 2 y + (- 3 y) \end{aligned}$ .
::简化 - 6 y + 2 y + ( - 3 y) 。

First, notice that all three terms have the same variable, $\begin{aligned} y \end{aligned}$ , so they are all like terms.
::首先,请注意所有三个术语都有相同的变量,y, 所以它们都像术语一样。

Now, you will do this problem in two steps. Start by adding $\begin{aligned} - 6 y + 2 y \end{aligned}$ .
::现在,你将分两步来解决这个问题。从添加 - 6 y + 2 y 开始。

Look at the signs of the terms. $\begin{aligned} - 6 y \end{aligned}$ is negative, but $\begin{aligned} 2 y \end{aligned}$ is positive. Because they have different signs, you will need to subtract their absolute values.
::看看术语的符号。 - 6 y 是负值, 但 2 y 是正值。因为它们有不同的符号, 您需要减去它们的绝对值。

$\begin{aligned} | - 6 | = 6 \end{aligned}$ and $\begin{aligned} | 2 | = 2 \end{aligned}$ .
::=6 =6 =2 =2 =2 =2 = = = = = = = = = = = = = = = = = = = = = = = = = = = = 6 = = 6 = = 6 = 2 = 2 = = = 2 = 2 = 2 = = = = = = = = = = = = = = = = = = = = = = = = = = = = 2 = 2 = = = 2 = = 2 = = 2 = = = 2 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 2 = 2 = 2 = 2 = = = = = = = = = = = 2 = 2 = 2 = 2 = 2 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 2 = = = = = = = = = 2 = = = = = = = = = = = = = = = = = = = = = =

$\begin{aligned} 6 - 2 = 4 \end{aligned}$

Your next step is to decide what the sign should be on your result. Your result should take the sign of the original term that had the greater absolute value. Since 6 is greater than 2, and $\begin{aligned} - 6 y \end{aligned}$ has a negative sign, give the result a negative sign.
::您的下一步是决定您的结果应该是什么牌子。您的结果应该带有具有更大绝对值的原始术语的牌子。由于 6 大于 2 和 - 6 y 有负的牌子, 请给结果一个负牌子。

$\begin{aligned} - 6 y + 2 y = - 4 y \end{aligned}$
::- 6y+2y=4y

Next, take $\begin{aligned} - 4 y \end{aligned}$ and add to it the final term from the original expression, $\begin{aligned} - 3 y \end{aligned}$ .
::接下来,取- 4 y, 并在它后面加上原表达式的最后一个任期, 3 y 。

Again, look at the signs of the terms. $\begin{aligned} - 4 y \end{aligned}$ is negative and $\begin{aligned} - 3 y \end{aligned}$ is also negative. Because they have the same sign, you will need to add their absolute values.
::再看看这些术语的符号。 4 Y 是负值, 3 y 也是负值。由于它们有相同的符号, 您需要添加它们的绝对值。

$\begin{aligned} | - 4 | = 4 \end{aligned}$ and $\begin{aligned} | - 3 | = 3 \end{aligned}$ .
::+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++3+3+3+3+3+3+3+3+3+3+3+3++++++3+++++++3+3+3++++++++++++++++++++++++++++3++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++3+3+++++3+3+3+3+3++++++++++++++++++++++++++3+++++++++++++++++++++++++++++++++++++++++++++++++++++++++

$\begin{aligned} 4 + 3 = 7 \end{aligned}$

Your next step is to decide what the sign should be on your final answer. Your answer should take the sign of the original terms. Since the original terms ( $\begin{aligned} - 4 y \end{aligned}$ and $\begin{aligned} - 3 y \end{aligned}$ ) were negative, your final answer should be negative.
::您的下一步是决定最后答案上应该写什么标志。您的答案应该写上原始条件的符号。由于原来的条件( 4 y 和 3 y) 是否定的, 您的最后答案应该是否定的。

$\begin{aligned} - 4 y + (- 3 y) = - 7 y \end{aligned}$
::- 4y+(-3y)=-7y

The answer is $\begin{aligned} - 6 y + 2 y + (- 3 y) = - 7 y \end{aligned}$ .
::答案是 - 6 y + 2 y + ( - 3 y) = 7 y 。

Review
::回顾

Simplify each variable expression.
::简化每个变量表达式。

$\begin{aligned} 7 z + (- 3 z) \end{aligned}$
::7z + (- 3z)

$\begin{aligned} 17 z + (- 15 z) \end{aligned}$
::17 z + (- 15 z)

$\begin{aligned} 5 x + (- 3 x) \end{aligned}$
::5 x + ( - 3 x ) 5 x + ( - 3 x )

$\begin{aligned} 8 y + (2 y) \end{aligned}$
::8 y + (2 y)

$\begin{aligned} 12 x + (- 13 x) \end{aligned}$
::12 x + (- 13 x )

$\begin{aligned} 9 z + (- 9 z) \end{aligned}$
::9 z + (- 9 z)

$\begin{aligned} 14 a + (- 3 a) \end{aligned}$
::14 a + (- 3 a)

$\begin{aligned} 22 y + (- 33 y) \end{aligned}$
::22 y + (- 33 y)

$\begin{aligned} (- 10 d) + (- d) + 2 \end{aligned}$
:-10 d)+(- d)+2

$\begin{aligned} 8 x + (- 4 x) - 5 \end{aligned}$
::8 x + ( - 4 x ) - 5 + ( - 4 x ) - 5

$\begin{aligned} 7 y + (- 3 y) \end{aligned}$
::7 y + (- 3 y)

$\begin{aligned} 16 x + (- 22 x) \end{aligned}$
::16 x + (- 22 x )

$\begin{aligned} 5 a + (- a) + 7 a \end{aligned}$
::5 a + (- a)+ 7 a

Solve each real-world problem.
::解决每个现实世界的问题

A plane is flying at an altitude that is 2, 500 feet above sea level. If the plane increases its altitude by 500 feet more, what will be its new altitude?
::飞机飞行高度为海平面2 500英尺以上,如果飞机的高度增加500英尺以上,新的高度是什么?

The temperature on a mountaintop at midnight was $\begin{aligned} - 8^{\circ} F \end{aligned}$ . By 3:00 A.M., the temperature had risen by $\begin{aligned} 3^{\circ} F \end{aligned}$ . What is the temperature at 3:00 A.M.?
::午夜的山顶温度是8°F 到凌晨3点温度上升了3°F

Review (Answers)
::回顾(答复)

Section outline

General

简化包含整数加法的变量表达式

Simplifying Variable Expressions Involving Integer Addition ::包含整数添加的简化变量表达式

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)

Simplifying Variable Expressions Involving Integer Addition
::包含整数添加的简化变量表达式

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)