Course: 2.9小数方程中加法的逆性质

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小数方程中加法的逆性质
Sean is saving up to buy a new video game. He keeps his money in a jar in his room. He knows that he currently has $28.91 saved up. Sean’s little sister Maria has just gotten interested in money. Sean thinks Maria might have taken some of his money out of the jar. He dumps out all the money in the jar and counts it up- only $25.74! How can Sean figure out how much money Maria took from the jar?
::Sean在存钱去买新的电子游戏。他把钱放在房间里的罐子里。他知道他现在存了28.91美元。 Sean的妹妹Maria刚刚对钱感兴趣。Sean认为Maria可能从罐子里拿走了部分钱。他把所有的钱都扔进罐里,数起来只有25.74美元!Sean怎么知道Maria从罐里拿了多少钱?

In this concept, you will learn how to apply the inverse property of addition to solve decimal equations.
::在此概念中,您将学会如何应用附加的反属性来解析小数方程式。

Applying the Inverse Property of Addition
::应用反向属性添加

In mathematics, inverse operations are operations that reverse one another. Addition and subtraction are inverse operations. For example, if you take any number and add 5 to it and then subtract 5 from the total, you will be back to the original number. The subtraction reversed the addition.
::在数学中,反向操作是相互反向的操作。增减是反向操作。例如,如果使用任何数字,加5,再加5,然后从总数中减去5,则返回到原来的数字。减法颠倒了增加值。

The Inverse Property of Addition states that the sum of any number and its opposite is zero. In symbols, it says that for any number $\begin{aligned} a \end{aligned}$ :
::反面加注财产称,任何数字之和及其相反数之和为零。在符号中,它表示,对于任何数字a:

$\begin{aligned} a + (- a) = 0 \end{aligned}$

The additive inverse of a number is another word for the opposite of a number. The additive inverse of $\begin{aligned} a \end{aligned}$ is $\begin{aligned} - a \end{aligned}$ .
::数字的添加反比是数字对面的另一个词。a的添加反比是-a。

You can use the inverse property of addition to help you to solve equations that would be difficult to solve using mental math. Remember that when you are solving an equation , your goal is to figure out the value of the variable that will make both sides of the equation equal.
::您可以使用附加的反向属性来帮助您解析使用心理数学难以解析的方程式。记住, 当您正在解析方程式时, 您的目标是要找出使方程式两侧均相等的变量值。

Here is an example.
::举一个例子。

Solve the following equation for $\begin{aligned} x \end{aligned}$ .
::解决 x 的以下方程式。

$\begin{aligned} x + 39.517 = 50.281 \end{aligned}$
::x + 39.517 = 50.281

First, notice that 39.517 is added to $\begin{aligned} x \end{aligned}$ on one side of the equation. You want to isolate the $\begin{aligned} x \end{aligned}$ which means you want to get the $\begin{aligned} x \end{aligned}$ by itself on one side of the equation. Subtract 39.517 from both sides of the equation.
::首先,请注意,39.517是在方程的一面添加到 x x 的。您想要分离 x , 这意味着您想要在方程的一面获得 x 。从方程的两面都减去 39.517 。

$\begin{aligned} x + 39.517 - 39.517 = 50.281 - 39.517 \end{aligned}$

Next, simplify the left side of the equation by combining like terms . $\begin{aligned} 39.517 - 39.517 \end{aligned}$ makes 0 which leaves the $\begin{aligned} x \end{aligned}$ by itself.
::接下来,简化方程的左侧,将类似条件合并。 39.517 - 39.517 make 0 使x 本身留下。

$\begin{aligned} x = 50.281 - 39.517 \end{aligned}$

Now, simplify the right side of the equation by combining like terms. Use what you have learned about decimal subtraction.
::现在, 简化方程的右侧, 将类似条件合并。使用您所学到的关于小数减法的知识。

$\begin{aligned} x = 10.764 \end{aligned}$

The answer is $\begin{aligned} x = 10.764 \end{aligned}$ .
::答案是x=10.764。

You can check your solution by substituting that value for $\begin{aligned} x \end{aligned}$ back into the original equation and verifying that it makes both sides equal.
::您可以检查您的解决方案, 将 x 的数值替换为返回原始方程, 并验证它是否使两边都相等。

$\begin{aligned} \begin{array}{rcl} x + 39.517 & = & 50.281 \\ 10.764 + 39.517 & = & 50.281 \\ 50.281 & = & 50.281 \end{array} \end{aligned}$

Your answer is correct.
::你的回答是对的

Remember that when solving equations using the inverse property of addition, you will always need to add or subtract the same amount from both sides of the equation in order to keep both sides equal to one another.
::记住,在用附加的反向属性解析方程式时,你总是需要从方程式的两侧增减相同数额,以使双方保持平等。

Here is another example.
::下面是另一个例子。

Solve the following equation for $\begin{aligned} x \end{aligned}$ .
::解决 x 的以下方程式。

$\begin{aligned} x - 43.27 = 182.205 \end{aligned}$

::x - 43.27 = 182.205

First, notice that 43.27 is subtracted from $\begin{aligned} x \end{aligned}$ on one side of the equation. To isolate $\begin{aligned} x \end{aligned}$ , add 43.27 to both sides of the equation.
::首先,请注意从方程式一边的 x 中减去43.27。要孤立 x,在方程式的两侧增加43.27。

$\begin{aligned} x - 43.27 + 43.27 = 182.205 + 43.27 \end{aligned}$

Next, simplify the left side of the equation by combining like terms. $\begin{aligned} - 43.27 + 43.27 \end{aligned}$ makes 0 which leaves the $\begin{aligned} x \end{aligned}$ by itself.
::接下来,简化方程的左侧,将类似术语合并。 - 43.27 + 43.27 使 0 使 x 本身留下。

$\begin{aligned} x = 182.205 + 43.27 \end{aligned}$

Now, simplify the right side of the equation by combining like terms. Use what you have learned about decimal addition.
::现在, 简化方程的右侧, 将类似条件合并。使用您所学到的关于小数增加的内容。

$\begin{aligned} x = 225.475 \end{aligned}$

The answer is $\begin{aligned} x = 225.475 \end{aligned}$ .
::答案是x=225.475。

Next, check your solution.
::下一个,检查你的解决方案。

$\begin{aligned} \begin{array}{rcl} x - 43.27 & = & 182.205 \\ 225.475 - 43.27 & = & 182.205 \\ 182.205 & = & 182.205 \end{array} \end{aligned}$

Your answer is correct.
::你的回答是对的

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Sean and his sister Maria.
::早些时候,你得到一个问题西恩和他的妹妹玛利亚。

Sean had $28.91 saved up in a jar in his room, but now the jar only has $25.74 in it. Sean wants to figure out how much money Maria took from the jar.
::Sean在房间里的罐子里存了28.91美元, 但现在罐子只有25.74美元。Sean想知道Maria从罐子里拿了多少钱。

First, define a variable. In this problem, you are trying to figure out the amount of money that Maria took from the jar.
::首先,定义一个变量。在这个问题上,你试图弄清楚 Maria从罐子上拿的钱的数量。

Let the variable $\begin{aligned} x \end{aligned}$ be equal to the amount of money Maria took from the jar.
::让变量 x 等于 Maria 从罐子中拿的钱。

Now, write an equation. You know that if you take the number of amount of money currently in the jar and add the amount Maria took, you will get the amount of money Sean started with. You know there is currently $25.74 in the jar and Sean started with $28.91.
::现在,写一个方程。你知道,如果你把目前罐子里的钱数量加进Maria的金额, 你会得到肖恩开始的钱数量。你知道,现在罐子里有25.74美元,肖恩从28.91美元开始。

$\begin{aligned} 25.74 + x = 28.91 \end{aligned}$

Finally, solve the equation. To isolate $\begin{aligned} x \end{aligned}$ , subtract 25.74 from both sides of the equation.
::最后,解开方程。要分离 x,从方程的两侧减去25.74。

$\begin{aligned} 25.74 + x - 25.74 = 28.91 - 25.74 \end{aligned}$

Next, simplify the left side of the equation by combining like terms. $\begin{aligned} 25.74 - 25.74 \end{aligned}$ makes 0, leaving the $\begin{aligned} x \end{aligned}$ by itself.
::接下来,简化方程的左侧,将类似条件合并。 25.74 - 25.74 makes 0, 将 x 自己留置。

$\begin{aligned} x = 28.91 - 25.74 \end{aligned}$

Now, simplify the right side of the equation by combining like terms.
::现在,简化方程的右侧, 将类似条件合并。

$\begin{aligned} x = 3.17 \end{aligned}$

The answer is Maria took $3.17 from the jar.
::答案是Maria从罐子里拿走了3.17美元

Example 2
::例2

At Saturday’s track meet, Liz ran 1.96 kilometers less than Sonya. Liz ran 1.258 kilometers. How many kilometers did Sonya run ?
::在星期六的赛马场上,Liz的跑程比Sonya少1.96公里。Liz的跑程比Sonya少了1.258公里。 Sonya的跑程是多少公里?

First, notice that the key words “less than” are a clue that this might be a subtraction equation.
::首先,请注意关键词“低于”是一个提示,它可能是减法方程。

Next, define a variable. In this problem, you are trying to figure out the number of kilometers Sonya ran.
::接下来,定义一个变量。在这个问题中, 您正在尝试找出 Sonya 所运行的公里数。

Let the variable $\begin{aligned} x \end{aligned}$ be equal to the number of kilometers Sonya ran.
::让变量 x 等于 Sonya 运行的公里数。

Now, write an equation. You know that if you take the number of kilometers Sonya ran and subtract 1.96, you will get the number of kilometers Liz ran. You also know that Liz ran 1.258 kilometers.
::现在,写一个方程。你知道,如果你把Sonya 跑的公里数减去1.96公里,你就会得到Liz 跑的公里数。你也知道Liz 跑的1.258公里。

$\begin{aligned} x - 1.96 = 1.258 \end{aligned}$

Finally, solve the equation. To isolate $\begin{aligned} x \end{aligned}$ , add 1.96 to both sides of the equation.
::最后,解开方程。要分离 x,在方程的两侧增加1.96。

$\begin{aligned} x - 1.96 + 1.96 = 1.258 + 1.96 \end{aligned}$

Next, simplify the left side of the equation by combining like terms.
::接下来,简化方程的左侧, 将类似条件合并。

$\begin{aligned} x = 1.258 + 1.96 \end{aligned}$

Now, simplify the right side of the equation by combining like terms.
::现在,简化方程的右侧, 将类似条件合并。

$\begin{aligned} x = 3.218 \end{aligned}$

The answer is Sonya ran 3.218 kilometers.
::答案是 Sonya跑3.218公里

Example 3
::例3

Solve the following equation for $\begin{aligned} x \end{aligned}$ .
::解决 x 的以下方程式。

$\begin{aligned} x + 5.678 = 12.765 \end{aligned}$
::x + 5.678 = 12.765

First, notice that 5.678 is added to $\begin{aligned} x \end{aligned}$ on one side of the equation. To isolate $\begin{aligned} x \end{aligned}$ , subtract 5.678 from both sides of the equation.
::首先,请注意方程一边的 x 中加上5.678。要分离 x,从方程两侧减去5.678。

$\begin{aligned} x + 5.678 - 5.678 = 12.765 - 5.678 \end{aligned}$

Next, simplify both sides of the equation by combining like terms.
::接下来,简化等式的两边,将类似条件结合起来。

$\begin{aligned} x = 7.087 \end{aligned}$

The answer is $\begin{aligned} x = 7.087 \end{aligned}$ .
::答案是 x = 7. 087。

Next, check your solution.
::下一个,检查你的解决方案。

$\begin{aligned} \begin{array}{rcl} x + 5.678 & = & 12.765 \\ 7.087 + 5.678 & = & 12.765 \\ 12.765 & = & 12.765 \end{array} \end{aligned}$

Your answer is correct.
::你的回答是对的

Example 4
::例4

Solve the following equation for $\begin{aligned} x \end{aligned}$ .
::解决 x 的以下方程式。

$\begin{aligned} x - 4.32 = 19.87 \end{aligned}$
::x - 4.32 = 19.87

First, notice that 4.32 is subtracted from $\begin{aligned} x \end{aligned}$ on one side of the equation. To isolate $\begin{aligned} x \end{aligned}$ , add 4.32 to both sides of the equation.
::首先,请注意从方程式一边的 x 中减去4.32。要孤立 x,在方程式的两侧增加4.32。

$\begin{aligned} x - 4.32 + 4.32 = 19.87 + 4.32 \end{aligned}$

Next, simplify both sides of the equation by combining like terms.
::接下来,简化等式的两边,将类似条件结合起来。

$\begin{aligned} x = 24.19 \end{aligned}$

The answer is $\begin{aligned} x = 24.19 \end{aligned}$ .
::答案是x=24.19。

Next, check your solution.
::下一个,检查你的解决方案。

$\begin{aligned} \begin{array}{rcl} x - 4.32 & = & 19.87 \\ 24.19 - 4.32 & = & 19.87 \\ 19.87 & = & 19.87 \end{array} \end{aligned}$

Your answer is correct.
::你的回答是对的

Example 5
::例5

Solve the following equation for $\begin{aligned} x \end{aligned}$ .
::解决 x 的以下方程式。

$\begin{aligned} x + 123.578 = 469.333 \end{aligned}$
::x + 123.578 = 469.333

First, notice that 123.578 is added to $\begin{aligned} x \end{aligned}$ on one side of the equation. To isolate $\begin{aligned} x \end{aligned}$ , subtract 123.578 from both sides of the equation.
::首先,请注意方程一面的 x 中加上 123.578。要分离 x,从方程两侧减去 123.578。

$\begin{aligned} x + 123.578 - 123.578 = 469.333 - 123.578 \end{aligned}$

Next, simplify both sides of the equation by combining like terms.
::接下来,简化等式的两边,将类似条件结合起来。

$\begin{aligned} x = 345.755 \end{aligned}$

The answer is $\begin{aligned} x = 345.755 \end{aligned}$ .
::答案是 x = 345.755。

Next, check your solution.
::下一个,检查你的解决方案。

$\begin{aligned} \begin{array}{rcl} x + 123.578 & = & 469.333 \\ 345.755 + 123.578 & = & 469.333 \\ 469.333 & = & 469.333 \end{array} \end{aligned}$

Your answer is correct.
::你的回答是对的

Review
::回顾

Solve each equation for $\begin{aligned} x \end{aligned}$ .
::为 x 解决每个方程式。

$\begin{aligned} x + 2.39 = 7.01 \end{aligned}$
::x + 2.39 = 7.01

$\begin{aligned} x + 5.64 = 17.22 \end{aligned}$
::x + 5.64 = 17.22

$\begin{aligned} x + 8.07 = 18.12 \end{aligned}$
::x + 8.07 = 18.12

$\begin{aligned} x + 14.39 = 17.342 \end{aligned}$
::x + 14.39 = 17.342

$\begin{aligned} x + 21.3 = 87.12 \end{aligned}$
::x + 21.3 = 87.12

$\begin{aligned} x + 31.9 = 77.22 \end{aligned}$
::x+31.9=77.22

$\begin{aligned} x + 18.77 = 97.12 \end{aligned}$
::x + 18.77 = 97.12

$\begin{aligned} x + 21.31 = 27.09 \end{aligned}$
::x + 21.31 = 27.09

$\begin{aligned} x + 18.11 = 87.22 \end{aligned}$
::x+18.11=87.22

$\begin{aligned} 818.703 = 614.208 + x \end{aligned}$
::818.703 = 614.208 + x

$\begin{aligned} x + 55.27 = 100.95 \end{aligned}$
::x + 55.27 = 100.95

Use equations to solve each word problem. Each answer should have an equation and a value for the variable.
::使用方程式解决每个单词问题。每个答案应该有一个方程式和变量的值。

Jamal’s leek and potato soup calls for 2.45 kg more potatoes than leeks. Jamal uses 4.05 kg of potatoes. How many kilograms of leeks does he use? Write an equation and solve.
::Jamal的利克和土豆汤要求的马铃薯比利克斯多2,45公斤。Jamal用的是4.05公斤土豆。他用了多少公斤的利克?写一个方程和解析。

The distance east from Waterville to Longford is 67.729 kilometers. The distance west from Waterville to Treetown is 61.234 kilometers. What is the difference between the two distances? Write an equation and solve.
::从沃特维尔到隆福东边的距离为67.729公里,从沃特维尔到树镇西边的距离为61.234公里。这两条距离有什么区别? 写一个方程和解答。

Sabrina spent $25.62 at the book fair. When she left the fair, she had $6.87. How much did money did she take to the fair? Write an equation and solve.
::Sabrina在书展上花了25.62美元。当她离开展销会时,她有6.87美元。她拿了多少钱去展销会?写一个方程式和解答。

Mr. Bodin has 11.25 liters of a cleaning solution, which is a combination of soap and water. If there are 2.75 liters of soap in the solution, how many liters of water are in the solution? Write an equation and solve.
::Bodin先生有11.25升清洁溶液,这是肥皂和水的结合,如果有2.75升肥皂在溶液中,有多少升水在溶液中?写一个方程式和溶液。

Review (Answers)
::回顾(答复)

Section outline

General

小数方程中加法的逆性质

Applying the Inverse Property of Addition ::应用反向属性添加

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)

Applying the Inverse Property of Addition
::应用反向属性添加

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)