Course: 1.11加减法方程的心算

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加减法方程的心算
Nico has a pet chihuahua named Daisy. Nico spends a lot of time carrying Daisy around so he started wondering how much Daisy weighs. Since Daisy is too squirmy and light to get on the scale herself, Nico comes up with another plan. First, he weighs himself alone on the scale and finds that he weighs 91 pounds. Then, he picks Daisy up and gets on the scale again and finds that he and Daisy together weigh 96 pounds. How can Nico figure out how much Daisy weighs?
::尼科有一个叫Daisy的宠物吉娃娃。尼科花了很多时间带着Daisy,所以他开始怀疑Daisy的体重有多重。由于Daisy的体重太小,光线太小,所以尼科又想出另一个计划。首先,他独自在体重上称自己,发现自己有91磅重。然后,他接了Daisy,又回到体重上,发现他和Daisy一起重96磅。尼科怎么知道Daisy的体重有多重?

In this concept, you will learn how to solve single variable addition and subtraction equations using mental math.
::在此概念中, 您将学习如何使用心理数学解决单变量添加和减法等式。

Solving Addition and Subtraction Equations Using Mental Math
::利用心理数学解决增加和减法等量

An equation is a mathematical sentence that says that two mathematical expressions are equal. An equal sign , "=", separates the two equal mathematical expressions and indicates that you are dealing with an equation.
::等式是一个数学句子, 它表示两个数学表达式是相等的。等号“ = ” 将两个等式数学表达式分开, 表示您正在处理一个等式。

Here are some examples of equations.
::以下是一些方程式的例子。

$\begin{aligned} 2 + 10 = 6 + 6 \end{aligned}$

$\begin{aligned} 11 - 8 = 1 + 1 + 1 \end{aligned}$

$\begin{aligned} x + 10 = 12 \end{aligned}$
::x + 10 = 12

$\begin{aligned} 2^{x} = 8 \end{aligned}$
::2x=8

A variable equation is an equation that contains a variable.
::变量方程式是一个包含变量的方程式。

Here are some examples of variable equations.
::以下是可变方程式的一些例子。

$\begin{aligned} 15 t = 45 \end{aligned}$
::15吨=45

$\begin{aligned} 12 = x + 9 \end{aligned}$
::12 = x + 9

$\begin{aligned} 6^{x} = 216 \end{aligned}$
::6x=216

$\begin{aligned} 25 = 3 (x - 7) + 1 \end{aligned}$
::25=3(x-7)+1

To solve a variable equation means to determine the value of the variable that makes the equation true. In other words, when you solve an equation you are looking for the value of the variable that makes both sides of the equation equal. This value of the variable that makes both sides of the equation equal is called the solution .
::解析变量方程式意味着确定使公式真实化的变量值。换句话说, 当解析公式时, 您正在寻找使方程式两侧均相等的变量值。使方程式两侧均等化的变量值被称为解决方案。

Some variable equations are simple enough that you are able to determine the solution to the equation in your head using mental math.
::一些变量方程式非常简单,您能够用心理数学来决定您脑中方程式的解决方案。

Let's look at an example that involves addition.
::让我们来举个例子,增加一些内容。

Solve the equation $\begin{aligned} y + 10 = 15 \end{aligned}$ .
::解决y+10=15的等式。

First, turn the equation into a question that you can ask yourself. Remember you are trying to figure out the value of $\begin{aligned} y \end{aligned}$ that will make both sides of the equation equal.
::首先, 将方程转换成您可以问自己的问题。记住, 您正在试图找出 y 的值, 使方程的两边都相等。

"What number plus 10 is equal to 15?"
::"什么数字加10等于15?"

Next, answer your question. You know that 5 plus 10 is equal to 15. This means $\begin{aligned} y \end{aligned}$ must be 5.
::接下来,回答你的问题。你知道5+10等于15,这意味着你必须是5。

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

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

Because your value for $\begin{aligned} y \end{aligned}$ results in both sides of the equation being equal, you can be confident that you have the correct answer. It's always a good idea to check your answer!
::因为你对y的价值导致等式的两面平等, 你可以相信你有正确的答案。检查你的答案总是个好主意!

Here is an example that involves subtraction.
::这里的例子涉及减法。

Solve the equation $\begin{aligned} x - 5 = 20 \end{aligned}$ .
::解决方程式 x - 5 = 20 。

First, turn the equation into a question that you can ask yourself.
::首先,把方程式变成一个问题你可以问自己。

"What number minus 5 is equal to 20?"
::"什么数减5等于20?"

Next, answer your question. You know that 25 minus 5 is equal to 20. This means $\begin{aligned} x \end{aligned}$ must be 25.
::接下来,回答你的问题。你知道25减5等于20,这意味着x必须是25。

The answer is $\begin{aligned} x = 25 \end{aligned}$ .
::答案是 x = 25 。

Now, check your solution.
::现在,检查你的解决方案。

The answer is correct.
::答案是正确的。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Nico and his pet chihuahua Daisy.
::早些时候,你得到一个问题关于尼科和他的宠物吉娃娃黛西。

Nico wants to know how much Daisy weighs. He weighed himself alone and found out that he weighs 91 pounds. Then he picked Daisy up and found out that together they weigh 96 pounds.
::尼科想知道Daisy的体重有多重。他独自称重自己,发现自己有91磅重。然后他接了Daisy,发现他们一起96磅重。

First, Nico can define his variable. He doesn't know how much Daisy weighs so he can let $\begin{aligned} x \end{aligned}$ be Daisy's weight.
::首先,尼科可以定义他的变数。他不知道黛西的体重有多重,所以让X成为黛西的重量。

Next, he can write a variable equation. He knows that his weight (91 pounds) plus Daisy's weight must equal 96 pounds.
::他知道他的体重(91磅)加上黛西的重量必须等于96磅

$\begin{aligned} 91 + x = 96 \end{aligned}$
::91+x=96

Then, he can turn the equation into a question that he can ask himself.
::然后,他可以把等式变成一个问题, 他自己可以问自己。

"91 plus what number is equal to 96?"
::"91加什么数字等于96?"

Next, he can answer his question. 91 plus 5 is equal to 96. So $\begin{aligned} x \end{aligned}$ must be equal to 5.
::接下来,他可以回答他的问题。 91+5等于96, 所以x必须等于5。

The answer is that Daisy weighs 5 pounds.
::答案是黛西重5磅

Example 2
::例2

There are 63 players in the symphony orchestra. 29 of these players are women. How many players in the orchestra are men? Write a variable equation and solve.
::在交响乐团里有63个玩家。其中29个是女性。乐队里有多少个玩家是男性?写一个变量方程和解答。

First, note that the problem tells you the number of women in the orchestra as well as the total number of players. You know that the number of men players plus the number of women players will equal the total number of players, so you can write the following equation.
::首先,请注意,这个问题可以告诉你乐队中女性的人数以及球员的总数。你知道,男球员的人数加上女球员的人数将等于球员的总数,这样你就可以写下以下的方程式。

$\begin{aligned} x + 29 = 63 \end{aligned}$
::x + 29 = 63

Now, turn the equation into a question that you can ask yourself.
::现在,把方程式变成一个问题你可以问自己。

"What number plus 29 is equal to 63?"
::"什么数字加29等于63?"

Next, answer your question. You know that 30 plus 29 is equal to 59, which is 4 less than what you want. So 34 plus 29 must be equal to 63. This means $\begin{aligned} x \end{aligned}$ must be 34.
::接下来,回答你的问题。你知道30加29等于59,这比你想要的少4倍。34加29必须等于63。这意味着x必须是34。

The answer is that 34 of the players in the orchestra are men.
::答案是乐队的34名球员是男人

Example 3
::例3

Use mental math to solve the following equation.
::使用心理数学解析以下方程式。

$\begin{aligned} x + 5 = 12 \end{aligned}$
::x+5=12x+5=12

First, turn the equation into a question that you can ask yourself.
::首先,把方程式变成一个问题你可以问自己。

"What number plus 5 is equal to 12?"
::"什么数字加5等于12?"

Next, answer your question. You know that 7 plus 5 is equal to 12. This means $\begin{aligned} x \end{aligned}$ must be 7.
::接下来,回答你的问题。你知道7+5等于12,这意味着x必须是7。

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

Now, check your solution.
::现在,检查你的解决方案。

The answer is correct.
::答案是正确的。

Example 4
::例4

Use mental math to solve the following equation.
::使用心理数学解析以下方程式。

$\begin{aligned} x - 3 = 18 \end{aligned}$
::x - 3 = 18

First, turn the equation into a question that you can ask yourself.
::首先,把方程式变成一个问题你可以问自己。

"What number minus 3 is equal to 18?"
::"什么数减去3等于18?"

Next, answer your question. You know 21 minus 3 is equal to 18. This means $\begin{aligned} x \end{aligned}$ must be 21.
::接下来,回答你的问题。你知道21减3等于18,这意味着x必须是21。

The answer is $\begin{aligned} x = 21 \end{aligned}$ .
::答案是 x = 21 。

Now, check your solution.
::现在,检查你的解决方案。

The answer is correct.
::答案是正确的。

Example 5
::例5

Use mental math to solve the following equation.
::使用心理数学解析以下方程式。

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

First, turn the equation into a question that you can ask yourself.
::首先,把方程式变成一个问题你可以问自己。

"6 minus what number is equal to 4?"
::"6减去什么数字等于4?"

Next, answer your question. You know that 6 minus 2 is equal to 4. This means $\begin{aligned} y \end{aligned}$ must be 2.
::接下来,回答你的问题。你知道,6减2等于4,这意味着你必须是2。

The answer is $\begin{aligned} y = 2 \end{aligned}$ .
::答案是y=2。

Now, check your solution.
::现在,检查你的解决方案。

The answer is correct.
::答案是正确的。

Review
::回顾

Use mental math to solve each addition or subtraction equation.
::使用心理数学解析每个添加或减法方程式。

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

$\begin{aligned} y + 5 = 10 \end{aligned}$
::y + 5 = 10

$\begin{aligned} a + 7 = 20 \end{aligned}$
::a + 7 = 20

$\begin{aligned} b + 8 = 13 \end{aligned}$
::b + 8 = 13

$\begin{aligned} z + 10 = 32 \end{aligned}$
::z z + 10 = 32

$\begin{aligned} s + 14 = 26 \end{aligned}$
::s + 14 = 26

$\begin{aligned} 4 + y = 8 \end{aligned}$
::4+y=8

$\begin{aligned} 6 + x = 21 \end{aligned}$
::6+x=21

$\begin{aligned} 17 + a = 23 \end{aligned}$
::17+a=23

$\begin{aligned} 18 + b = 30 \end{aligned}$
::18 + b = 30

$\begin{aligned} 12 + x = 24 \end{aligned}$
::12+x=24

$\begin{aligned} 13 + y = 18 \end{aligned}$
::13+y=18岁

$\begin{aligned} 15 + a = 22 \end{aligned}$
::15 + a = 22

$\begin{aligned} x + 17 = 24 \end{aligned}$
::x+17=24x+17=24

$\begin{aligned} y + 3 = 45 \end{aligned}$
::y + 3 = 45

$\begin{aligned} x - 4 = 10 \end{aligned}$
::x - 4 = 10

$\begin{aligned} y - 8 = 20 \end{aligned}$
::y - 8 = 20

$\begin{aligned} 5 - y = 2 \end{aligned}$
::5-y = 2

$\begin{aligned} 22 - a = 15 \end{aligned}$
::22-a = 15

$\begin{aligned} 18 - y = 2 \end{aligned}$
::18-y = 2

Review (Answers)
::回顾(答复)

Section outline

General

加减法方程的心算

Solving Addition and Subtraction Equations Using Mental Math ::利用心理数学解决增加和减法等量

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)

Solving Addition and Subtraction Equations Using Mental Math
::利用心理数学解决增加和减法等量

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)