Course: 4.9简化包含整数减法的变量表达式

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简化包含整数减法的变量表达式
Noah has started up a lawn mowing business to make some extra money this summer. He charges $14 to mow a lawn. For each lawn he mows, it costs him $1.50 for supplies. If $\begin{aligned} L \end{aligned}$ stands for the number of lawns Noah mows, how could Noah write and simplify an expression for his profit from mowing lawns this summer? If Noah ends up mowing 130 lawns, how much money does he make this summer?
::诺亚今年夏天开始做草坪修剪生意,赚点额外的钱。他要14美元修剪草坪。每次修剪草坪都要花1.50美元。如果我是修剪草坪的诺亚修剪草坪的数量,那么诺亚今年夏天怎么写和简化他从修剪草坪中获利的表达方式呢?如果诺亚最后修剪130个草坪,那么他今年夏天赚多少钱呢?

In this concept, you will learn how to simplify and evaluate variable expressions involving integer subtraction .
::在此概念中,您将学会如何简化和评估涉及整数减法的变量表达式。

Simplifying Variable Expressions Involving Integer Subtraction
::涉及整数减法的简化变量表达式

A variable expression is a math phrase that has numbers, variables, and operations in it.
::变量表达式是一个数学短语,其中含有数字、变量和操作。

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

$\begin{aligned} - 2 x - x \end{aligned}$
::- 2 x - xx - x

$\begin{aligned} 3 x - 5 y \end{aligned}$
::3 x - 5 y

$\begin{aligned} - 10 n - (- 8 n) \end{aligned}$
::- 10n-(-8n)

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。

If two like terms are being subtracted in a variable expression, you can use what you know about integer subtraction to help you to combine the terms.
::如果在变量表达式中减去两个类似术语,您可以使用您所了解的整数减法来帮助您合并术语。

Here is an example.
::举一个例子。

Simplify $\begin{aligned} - 10 n - (- 8 n) \end{aligned}$ .
::简化-10n-(-8n)

In this variable expression, $\begin{aligned} - 10 n \end{aligned}$ and $\begin{aligned} - 8 n \end{aligned}$ are like terms because they include the same variable, $\begin{aligned} n \end{aligned}$ .
::在这个变数表达式中,- 10 n和- 8 n是相似的术语,因为它们包括相同的变量 n。

First, rewrite this problem as an addition problem. The term being subtracted is $\begin{aligned} - 8 n \end{aligned}$ and the opposite of -8 is 8. This means subtracting $\begin{aligned} - 8 n \end{aligned}$ is the same as adding $\begin{aligned} 8 n \end{aligned}$ .
::首先,将这一问题重新写成一个附加问题。减去的术语是- 8 n, 与 - 8 的相反是- 8 。这意味着减去 - 8 n 与增加 8 n 相同。

$\begin{aligned} - 10 n - (- 8 n) = - 10 n + 8 n \end{aligned}$
::- 10n-(-8n)=-10n+8n

Now, notice that the two like terms have different signs. So, your next step is to find the absolute values of both integers and subtract.
::现在,请注意这两个类似术语有不同的符号。所以,你的下一步是找到两个整数和减数的绝对值。

$\begin{aligned} | - 10 | = 10 \end{aligned}$ and $\begin{aligned} | 8 | = 8 \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 8 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\begin{aligned} 10 - 8 = 2 \end{aligned}$

Then, 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 10 is greater than 8, and $\begin{aligned} - 10 n \end{aligned}$ has a negative sign, give the answer a negative sign.
::然后,请决定最后答案上应该写什么符号。您的答案应该写上具有更大绝对值的原始术语的符号。因为10岁大于8岁, 而- 10 n 有否定的符号, 请给答案一个否定的符号。

$\begin{aligned} - 10 n + 8 n = - 2 n \end{aligned}$
::- 10 n+8 n=-2 n

The answer is $\begin{aligned} - 10 n - (- 8 n) = - 2 n \end{aligned}$ .
::答案是-10n-(-8n)=-2n。

You can evaluate variable expressions for specific values. To evaluate means to determine what value the expression takes when the variable is equal to a specific number.
::您可以对特定值的变量表达式进行评价。要评价当变量等于一个特定数字时该表达式要使用何种手段来确定该表达式的值。

Here is an example.
::举一个例子。

Find $\begin{aligned} - 10 n - (- 8 n) \end{aligned}$ when $\begin{aligned} n = 5 \end{aligned}$ .
::查找-10n-(-8n)n=5。

You could evaluate by substituting $\begin{aligned} n = 5 \end{aligned}$ for both $\begin{aligned} n \end{aligned}$ 's in the expression and simplifying. However, you already know from the previous example that $\begin{aligned} - 10 n - (- 8 n) \end{aligned}$ simplifies to $\begin{aligned} - 2 n \end{aligned}$ . This means you can evaluate $\begin{aligned} - 2 n \end{aligned}$ for $\begin{aligned} n = 5 \end{aligned}$ instead of having to evaluate the original expression.
::您可以用 n = 5来替换表达式中的 n 和简化。但是,您已经从上一个例子中知道- 10 n = (- 8 n) 简化为-2 n。这意味着您可以对 n = 5 的评价- 2 n, 而不是要对原来的表达式进行评价。

$\begin{aligned} - 2 n = - 2 (5) = - 10 \end{aligned}$
::- 2 n = − 2 (5 ) = - 10

The answer is -10.
::答案是 -10

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Noah and his business mowing lawns.
::早些时候,你被问及诺亚和他的生意割草草坪的问题。

He charges $14 to mow a lawn, but every lawn he mows costs him $1.50 in supplies. First, he wanted to know an expression that would represent his profit in terms of $\begin{aligned} L \end{aligned}$ , the number of lawns he ends up mowing.
::他收取14美元修剪草坪,但每次修剪草坪都要花1.50美元的供应费。首先,他想知道一种表达方式,即他从L(L)中获利,他最终修剪的草坪数量。

He gets $\begin{aligned} 14 L \end{aligned}$ dollars from mowing $\begin{aligned} L \end{aligned}$ lawns, but he spends $\begin{aligned} 1.50 L \end{aligned}$ dollars in supplies from mowing those same lawns. That means his profit is
::他从修剪L草坪得到14里拉的14美元,但他从修剪这些草坪上花费1.50里拉的供给。这意味着他的利润就是他的利润。

$\begin{aligned} 14 L - 1.50 L \end{aligned}$
::14 L - 1.50 L

Now Noah can simplify this expression.
::现在Noah可以简化这个表达式。

Notice that both terms have the same variable, $\begin{aligned} L \end{aligned}$ , so they are like terms. You can rewrite the subtraction problem as an addition problem. The term being subtracted is $\begin{aligned} 1.50 L \end{aligned}$ . The opposite of 1.50 is -1.50. So you have
::注意这两个词都有相同的变量, L, 所以它们都是一样的。您可以将减法问题重写为添加问题。减法为 1. 50 L。与 1. 50 相对的是 - 1. 50 。因此, 您已经将减法问题重写为添加问题。

$\begin{aligned} 14 L - 1.50 L = 14 L + (- 1.50 L) \end{aligned}$
::14 L - 1.50 L = 14 L + ( - 1.50 L)

Now notice that the two like terms have different signs. Combine by subtracting their absolute values.
::现在请注意,这两个相似的术语有不同的符号。合并后减去其绝对值。

$\begin{aligned} | 14 | = 14 \end{aligned}$ and $\begin{aligned} | - 1.50 | = 1.50 \end{aligned}$
::14 = 14 = 14 和 = 1.50 = 1.50

$\begin{aligned} 14 - 1.50 = 12.50 \end{aligned}$

The result will have the same sign as the original term with the greater absolute value. Because $\begin{aligned} 14 L \end{aligned}$ has the greater absolute value, the result is positive.
::结果将具有与原术语相同的信号,其绝对值更大。因为14 L 具有更大的绝对值,结果为正值。

$\begin{aligned} 14 L + (- 1.50 L) = 12.50 L \end{aligned}$
::14 L+ (- 1.50 L) = 12.50 L

The simplified expression is $\begin{aligned} 14 L - 1.50 L = 12.50 L \end{aligned}$ .
::简化表达式为14 L - 1.50 L = 12.50 L 。

Noah also wondered what his profit would be if he ended up mowing 130 lawns during the summer. To figure this out, you can evaluate $\begin{aligned} 12.50 L \end{aligned}$ for $\begin{aligned} L = 130 \end{aligned}$ .
::诺亚也想知道,如果他在夏天修剪了130片草坪,他的利润会有多大。要弄清楚这一点,你可以对L=130进行12.50里拉的评估。

$\begin{aligned} 12.50 L = 12.50 (130) = 1625 \end{aligned}$ .
::12.50 L=12.50(130)=1625。

If Noah mowed 130 lawns, his profit would be $1625.
::如果诺亚割下130片草坪他的利润将是1625美元

Example 2
::例2

Simplify $\begin{aligned} - 33 b - (- 18 b) + 7 \end{aligned}$ .
::简化 - 33 b - ( - 18 b) + 7 。

First, notice that there are three terms. The first two terms are like terms because they have the same variable, $\begin{aligned} b \end{aligned}$ . Focus on combining the like terms: $\begin{aligned} - 33 b - (- 18 b) \end{aligned}$ .
::首先,请注意有三个术语。前两个术语相似,因为它们具有相同的变量, b。侧重于将类似术语合并:- 33 b- (- 18 b) 。

Next, rewrite this problem as an addition problem. The term being subtracted is $\begin{aligned} - 18 b \end{aligned}$ and the opposite of -18 is 18. This means subtracting $\begin{aligned} - 18 b \end{aligned}$ is the same as adding 18b.
::其次,将这一问题重新写成一个添加问题。要减去的术语是 - 18 b, 与 - 18 的相反是 - 18 。这意味着减去 - 18 b 与增加 18 b 相同。

$\begin{aligned} - 33 b - (- 18 b) = - 33 b + 18 b \end{aligned}$
::- 33 b - ( - 18 b) = - 33 b + 18 b

Now, notice that the two like terms have different signs. Find the absolute values of both integers and subtract.
::现在请注意, 两个相似的术语有不同的符号。查找整数和减数的绝对值。

$\begin{aligned} | - 33 | = 33 \end{aligned}$ and $\begin{aligned} | 18 | = 18 \end{aligned}$
::=33 =18 =18 =18

$\begin{aligned} 33 - 18 = 15 \end{aligned}$

Next decide what the sign should be on your combined term. Your answer should take the sign of the original term that had the greater absolute value. Since $\begin{aligned} - 33 b \end{aligned}$ has the greater absolute value, your combined term will be negative.
::下一步决定您合并任期的标志应该是什么。您的回答应该使用具有较大绝对值的原始术语的符号。由于 - 33 b 具有更大的绝对值, 您的合并任期将是负值。

$\begin{aligned} - 33 b - (- 18 b) = - 15 b \end{aligned}$
::- 33 b - ( - 18 b) = - 15 b

Now you can write your final answer. Don't forget to include the +7 from the original expression. 7 is not like terms with $\begin{aligned} - 15 b \end{aligned}$ because it does not have a " $\begin{aligned} b \end{aligned}$ ", so it cannot be combined with $\begin{aligned} - 15 b \end{aligned}$ .
::现在您可以写下最后答案。请不要忘记将原来的表达式中的+7 包含在内。 7 与 -15 b 的条件不同, 因为它没有 " b " , 所以不能与 -15 b 合并。

The answer is $\begin{aligned} - 15 b + 7 \end{aligned}$ .
::答案是 - 15 b+7 。

Example 3
::例3

Simplify $\begin{aligned} - 4 y - 6 y \end{aligned}$ and then evaluate for $\begin{aligned} y = 2 \end{aligned}$ .
::简化 - 4 y - 6 y, 然后对 y = 2 进行评价。

First, notice that both terms have the same variable, $\begin{aligned} y \end{aligned}$ , so they are like terms. You will combine the like terms and then evaluate.
::首先,请注意这两个词的变量是相同的,y,所以它们就像术语一样。你将把相似的术语组合起来,然后进行评估。

Next, rewrite the subtraction problem as an addition problem. The term being subtracted is $\begin{aligned} 6 y \end{aligned}$ . The opposite of 6 is -6. So you have
::下一步,将减法问题重写为添加问题。要减去的术语是 6 y 。对应于 6 是 - 6。所以您有

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

Now notice that the two like terms have the same sign. Combine by adding their absolute values.
::现在请注意,这两个相似的术语有相同的符号。合并后添加其绝对值。

$\begin{aligned} | - 4 | = 4 \end{aligned}$ and $\begin{aligned} | - 6 | = 6 \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\begin{aligned} 4 + 6 = 10 \end{aligned}$

The result will have the same sign as the original two terms. Because the original terms were negative, the result is negative.
::结果将具有与最初两个术语相同的标志。因为最初的术语是否定的,结果就是否定的。

$\begin{aligned} - 4 y + (- 6 y) = - 10 y \end{aligned}$
::- 4y+(-6y)=-10y

The simplified expression is $\begin{aligned} - 4 y - 6 y = - 10 y \end{aligned}$ .
::简化表达式为 - 4 y - 6 y = - 10 y 。

Finally, you can evaluate $\begin{aligned} - 10 y \end{aligned}$ for $\begin{aligned} y = 2 \end{aligned}$ .
::最后,你可以对y=2进行10y的评分。

$\begin{aligned} - 10 y = - 10 (2) = - 20 \end{aligned}$
::- 10 y = - 10 (2) = - 20

The answer is -20.
::答案是 -20

Example 4
::例4

Simplify $\begin{aligned} 18 x - (- 4 x) \end{aligned}$ .
::简化 18 x - ( - 4 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, rewrite the subtraction problem as an addition problem. The term being subtracted is $\begin{aligned} - 4 x \end{aligned}$ . The opposite of -4 is 4. So you have
::下一步,将减法问题重写为添加问题。要减去的术语为 - 4 x 。与 - 4 相反的是 4。所以您有

$\begin{aligned} 18 x - (- 4 x) = 18 x + 4 x \end{aligned}$
::18x- (- 4x) = 18x + 4x = 18x + 4x

Now notice that the two like terms have the same sign. Combine by adding their absolute values.
::现在请注意,这两个相似的术语有相同的符号。合并后添加其绝对值。

$\begin{aligned} | 18 | = 18 \end{aligned}$ and $\begin{aligned} | 4 | = 4 \end{aligned}$
::18 = 18 = 18 = 4 = 4 = 4

$\begin{aligned} 18 + 4 = 22 \end{aligned}$

The result will have the same sign as the original two terms. Because the original terms were positive, the result is positive.
::结果将具有与最初两个术语相同的标志。因为最初的术语是正的,结果就是正的。

$\begin{aligned} 18 x + 4 x = 22 x \end{aligned}$
::18 x + 4 x = 22 x 18 x + 4 x = 22 x

The answer is $\begin{aligned} 18 x - (- 4 x) = 22 x \end{aligned}$ .
::答案是18x-(-4x)=22x。

Example 5
::例5

Simplify $\begin{aligned} - 9 a - (- 3 a) \end{aligned}$ and then evaluate for $\begin{aligned} a = 2 \end{aligned}$ .
::简化- 9 a - ( - 3 a) ,然后对 = 2 进行评价。

First, notice that both terms have the same variable, $\begin{aligned} a \end{aligned}$ , so they are like terms. You will combine the like terms and then evaluate.
::首先,请注意这两个词都有相同的变量, a, 所以它们就像术语一样。您可以将相似的术语组合起来, 然后评估。

Next, rewrite the subtraction problem as an addition problem. The term being subtracted is $\begin{aligned} - 3 a \end{aligned}$ . The opposite of -3 is 3. So you have
::下一步,将减法问题重写为添加问题。要减去的术语是 - 3 a。与 - 3 相反的是 3。所以您有

$\begin{aligned} - 9 a - (- 3 a) = - 9 a + 3 a \end{aligned}$
::- 9 a - (- 3 a) = - 9 a+ 3 a) = - 9 a+ 3 a

Now notice that the two like terms have different signs. Combine by subtracting their absolute values.
::现在请注意,这两个相似的术语有不同的符号。合并后减去其绝对值。

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

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

The result will have the same sign as the original term with the greater absolute value. Because $\begin{aligned} - 9 a \end{aligned}$ has the greater absolute value, the result is negative.
::结果将具有与原术语相同的符号,其绝对值更大。因为- 9 a 具有更大的绝对值,结果为负值。

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

The simplified expression is $\begin{aligned} - 9 a - (- 3 a) = - 6 a \end{aligned}$ .
::简化表达式为-9 a-(-3 a) =-6 a。

Finally, you can evaluate $\begin{aligned} - 6 a \end{aligned}$ for $\begin{aligned} a = 2 \end{aligned}$ .
::最后,您可以评估 - 6 a = 2 。

$\begin{aligned} - 6 a = - 6 (2) = - 12. \end{aligned}$
::- 6 a = - 6 (2) = 12。

The answer is -12.
::答案是 -12

Review
::回顾

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

$\begin{aligned} - 8 m - 3 m \end{aligned}$
::- 8米-3米

$\begin{aligned} (- 7 c) - (- c) \end{aligned}$
:-7c)-(-c)

$\begin{aligned} - 19 a - (- 4 a) \end{aligned}$
::- 19 a - ( - 4 a)

$\begin{aligned} 12 a - (- 4 a) \end{aligned}$
::12 a - ( - 4 a)

$\begin{aligned} 6 a - (4 a) \end{aligned}$
::6 a - (4 a)

$\begin{aligned} 19 a - (24 a) \end{aligned}$
::19 a - (24 a)

$\begin{aligned} 13 z - (- 4 z) \end{aligned}$
::13 z- ( - 4 z)

$\begin{aligned} - 20 x - (14 x) \end{aligned}$
::- 20x-(14x)

$\begin{aligned} - 19 a - (18 a) \end{aligned}$
::- 19a - (18a)

$\begin{aligned} 56 x - 22 x \end{aligned}$
::56 x - 22 x

$\begin{aligned} 34 y - (- 6 y) \end{aligned}$
::34 y - ( - 6 y)

$\begin{aligned} 88 z - (- 44 x) \end{aligned}$
::88z-(-44x)

$\begin{aligned} - 19 a - (- 4 a) - 8 x \end{aligned}$
::- 19 a - (- 4 a) - 8 x

$\begin{aligned} - 19 a - (- 4 a) - 8 x - 2 \end{aligned}$
::- 19 a - (- 4 a) - 8 x - 2

$\begin{aligned} - 19 a - (- 4 a) - 5 y - 2 y \end{aligned}$
::- 19 a - ( - 4 a) - 5 y - 2 y

Review (Answers)
::回顾(答复)

Section outline

General

简化包含整数减法的变量表达式

Simplifying Variable Expressions Involving Integer Subtraction ::涉及整数减法的简化变量表达式

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 Subtraction
::涉及整数减法的简化变量表达式

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)