课程： 12.5单变量加法方程

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选择节单变量加法方程

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单变量加法方程
John’s soccer team is raising money to attend the two-day camp. The team’s goal is to raise the $450 camp registration fee by washing cars. At the end of the day, the team has made $380. The team decides to split the $450 registration cost equally, with each player responsible for $45. They also split the $380 equally, so each receives $38 toward their total cost. Now, John and the team need to know how much more each team member needs to add to the $38 to reach its goal.
::约翰的足球队为参加为期两天的营地募集资金。球队的目标是通过洗车提高450美元的营地注册费。归根结底,球队已经赚到了380美元。球队决定平分450美元的注册费,每个球员负责45美元。他们也平分380美元,因此每人得到38美元的总成本。现在,约翰和球队需要知道每个球队成员需要加多少38美元才能达到目标。

In this concept, you will learn to solve single- variable addition equation .
::在此概念中,您将学会解决单变量加法等式。

Solving Single-Variable Addition Equations
::解决单变式添加等同

To solve a single- variable equation means figuring out the value of the variable or the unknown number. There are a number of different ways to do this. The first method is to use mental math.
::要解决一个单变量方程式, 意味着要找出变量或未知数字的值。有多种不同的方法可以做到这一点。第一种方法是使用心理数学。

Let’s look at an example.
::让我们举个例子。

$\begin{aligned} 5 + x = 10 \end{aligned}$

Using mental math you can ask yourself, “Five plus what number is equal to 10?”
::使用心理数学,你可以问自己,“五加五等于十?”

The answer is 5.
::答案是5。

You can check your answer too. To do this, simply substitute the value for $\begin{aligned} x \end{aligned}$ into the equation and see if it forms a true statement.
::您也可以检查您的答案。要做到这一点, 只需在方程中将 x 的值替换为 x, 并查看它是否构成一个真实的语句。

$\begin{aligned} \begin{array}{rcl} 5 + x & = & 10 \\ 5 + 5 & = & 10 \\ 10 & = & 10 \end{array} \end{aligned}$

This is a true statement, so the answer is correct.
::这是一个真实的陈述,因此答案是正确的。

Sometimes, it may be more difficult to figure out the value of the variable using mental math. The second way of solving a single-variable equation involves using the inverse operation . An inverse operation is the opposite of the given operation .
::有时,用心理数学来算出变量的价值可能比较困难。解决单一变量等式的第二种方式是使用反向操作。反向操作与给定操作相反。

Here is an equation.
::这里有一个方程式。

$\begin{aligned} x + 27 = 43 \end{aligned}$

In the equation, the given operation is addition, so you can use the opposite of addition (subtraction) to solve the problem. You need to get the variable by itself to figure out its value and you use subtraction to do this.
::在方程中,给定的操作是附加的,所以您可以使用反向的加法(减法)来解决问题。您需要将变量自己弄清楚其值,然后使用减法来做到这一点。

First, identify the number that is being added to the variable, in this case, 27.
::首先,确定正在添加到变量中的数字,在此情况下为27。

Next, subtract 27 from both sides of the equation. That will leave the variable on one side of the equation with 0 and 16 on the other side. Remember, any number or variable plus 0 is the number or variable.
::下一步, 从方程的两侧减去 27 。这将在方程的一边留下变量, 在另一侧留下 0 和 16 。记住, 数字或变量加 0 是数字或变量。

$\begin{aligned} \begin{array}{rcl} x + 27 & = & 43 \\ - 27 & - 27 \\ x + 0 & = & 16 \\ x & = & 16 \end{array} \end{aligned}$

The answer is $\begin{aligned} x = 16 \end{aligned}$ .
::答案是 x = 16 。

Check your work by substituting the value for $\begin{aligned} x \end{aligned}$ back into the original equation. If it is true, then one side of the equation will equal the other side.
::检查您的工作, 将 x 的值替换为返回原始方程式。如果它是真实的, 那么方程式的一面将等于另一面。

$\begin{aligned} \begin{array}{rcl} x + 27 & = & 43 \\ 16 + 27 & = & 43 \\ 43 & = & 43 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Here is another equation.
::这是另一个方程式。

$\begin{aligned} 45 + x = 67 \end{aligned}$

Even though $\begin{aligned} x \end{aligned}$ is in the middle of the equation and not at the start of it, you can still use an inverse to sort it out.
::即使x在方程中间,而不是在方程开始时,你仍然可以用反方向来解析它。

The goal is to get the $\begin{aligned} x \end{aligned}$ alone on one side of the equal sign.
::目标是将x单独放在平等标志的一边。

First, identify the number being added to the variable, in this case, 45.
::首先,确定变量中增加的数值,在本案中为45。

Next, using the inverse of a positive 45, subtract 45 from both sides of the equation.
::接下来,使用正45的反面, 减去方程两侧的45。

$\begin{aligned} \begin{array}{rcl} 45 + x & = & 67 \\ - 45 & - 45 \\ x & = & 22 \end{array} \end{aligned}$

Notice again that $\begin{aligned} 45 - 45 \end{aligned}$ is equal to 0. So, the variable is by itself on the left side of the equals. On the right side, you subtracted 45 and got an answer of 22.
::请注意 45 - 45 等于 0 。因此变量本身在等值的左侧。在右侧, 您减去了 45, 并得到了 22 的回答。

Check your work by substituting 22 for $\begin{aligned} x \end{aligned}$ in the original equation. One side of the equation should equal the other side.
::检查您的工作, 在原始方程中以 x 22 替换 x 。方程的一面应该等于另一方。

$\begin{aligned} \begin{array}{rcl} 45 + x & = & 67 \\ 45 + 22 & = & 67 \\ 67 & = & 67 \end{array} \end{aligned}$

The answer checks out.
::答案检查出来。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about John and the soccer’s team’s car wash fundraiser.
::更早一些时候, John和足球队的汽车洗涤募捐活动给大家一个问题。

Each of the ten team members has $38, and needs to pay an equal percentage of the remaining balance. They figure that each team member will be responsible for is $45. How can John figure out how much more each team member needs to pay?
::十名小组成员每人38美元,需要支付相等比例的剩余余额。他们认为每个小组成员要负责45美元。约翰如何确定每个小组成员需要支付多少?

First, begin by using the information to write a phrase.
::首先,首先利用信息来写一个短语。

38 and an unknown number is 45
::38个,未知号码45个

Next, write the phrase as an expression
::下一步,将该短语写成表达式

$\begin{aligned} 38 + x = 45 \end{aligned}$

Then, identify the number that is being added to the variable $\begin{aligned} x \end{aligned}$ , in this case, 38.
::然后,在变量 x 中标明加入的数值,在此情况下为38。

Finally, using the inverse of 38, subtract 38 from both sides of the equation.
::最后,使用38的逆数,从方程的两侧减去38。

$\begin{aligned} \begin{array}{rcl} 38 + x & = & 45 \\ - 38 & = & - 38 \\ 0 + x & = & 7 \\ x & = & 7 \end{array} \end{aligned}$

Now, check the answer by plugging 7 in the equation for $\begin{aligned} x \end{aligned}$ .
::现在,检查答案,在 X 的方程式中插入 7 。

$\begin{aligned} \begin{array}{rcl} 38 + x & = & 45 \\ 38 + 7 & = & 45 \\ 45 & = & 45 \end{array} \end{aligned}$

The answer checks out.
::答案检查出来。

Each team member will need to pay an additional $7 to cover the registration fee.
::每个小组成员将需要另外支付7美元,以支付登记费。

Solve the following equations then check your answer. Write your answers in the form:
::解决以下方程式, 然后检查您的答案。以窗体形式写入您的答案 :

$\begin{aligned} Variable = \underline{} . \end{aligned}$
::变数=_。

Example 2
::例2

$\begin{aligned} x + 15 = 32 \end{aligned}$

First, identify the number that is being added to the variable, in this case, 15.
::首先,确定变量中添加的数值,在此情况下为15。

Next, using the inverse of 15, subtract 15 from both sides of the equation.
::其次,使用15的反数,从方程的两侧减去15。

$\begin{aligned} \begin{array}{rcl} x + 15 & = & 32 \\ - 15 & - 15 \\ x + 0 & = & 17 \end{array} \end{aligned}$

The answer is $\begin{aligned} x = 17 \end{aligned}$ .
::答案是 x = 17 。

Finally, check your answer by plugging 17 in the equation for $\begin{aligned} x \end{aligned}$ .
::最后,通过在 X 方程式中插入 17 来检查您的答复。

$\begin{aligned} \begin{array}{rcl} x + 15 & = & 32 \\ 17 + 15 & = & 32 \\ 32 & = & 32 \end{array} \end{aligned}$

The answer checks out.
::答案检查出来。

Example 3
::例3

$\begin{aligned} x + 16 = 22 \end{aligned}$

First, identify the number that is being added to the variable, in this case, 16.
::首先,确定变量中增加的数值,在本案中为16。

Next, using the inverse of 16, subtract 16 from both sides of the equation.
::其次,使用 16 的逆数,从方程的两侧减去 16 。

$\begin{aligned} \begin{array}{rcl} x + 16 & = & 22 \\ - 16 & - 16 \\ x + 0 & = & 6 \end{array} \end{aligned}$

The answer is $\begin{aligned} x = 6 \end{aligned}$ .
::答案是 x = 6 。

Finally, check your answer by plugging 6 in the equation for $\begin{aligned} x \end{aligned}$ .
::最后,通过在 X 方程式中插入 6 来检查您的答复。

$\begin{aligned} \begin{array}{rcl} 6 + 16 & = & 22 \\ 6 + 16 & = & 22 \\ 22 & = & 22 \end{array} \end{aligned}$

The answer checks out.
::答案检查出来。

Example 4
::例4

$\begin{aligned} y + 15 = 30 \end{aligned}$

First, identify the number that is being added to the variable, in this case, 15.
::首先,确定变量中添加的数值,在此情况下为15。

Next, using the inverse of 15, subtract 15 from both sides of the equation.
::其次,使用15的反数,从方程的两侧减去15。

$\begin{aligned} \begin{array}{rcl} y + 15 & = & 30 \\ - 15 & - 15 \\ y + 0 & = & 15 \end{array} \end{aligned}$

The answer is $\begin{aligned} y = 15 \end{aligned}$ .
::答案是y=15。

Finally, check your answer by plugging 15 in the equation for $\begin{aligned} y \end{aligned}$ .
::最后,通过在y的方程中插入 15 来检查你的答案。

$\begin{aligned} \begin{array}{rcl} y + 15 & = & 30 \\ 15 + 15 & = & 30 \\ 30 & = & 30 \end{array} \end{aligned}$

The answer checks out.
::答案检查出来。

Example 5
::例5

$\begin{aligned} 12 + x = 18 \end{aligned}$

First, identify the number that is being added to the variable, in this case, 12.
::首先,确定变量中添加的数值,在本案中为12。

Next, using the inverse of 15, subtract 15 from both sides of the equation.
::其次,使用15的反数,从方程的两侧减去15。

$\begin{aligned} \begin{array}{rcl} x + 12 & = & 18 \\ - 12 & - 12 \\ x + 0 & = & 6 \end{array} \end{aligned}$

The answer is $\begin{aligned} x = 6 \end{aligned}$ .
::答案是 x = 6 。

Finally, check your answer by plugging 6 in the equation for $\begin{aligned} x \end{aligned}$ .
::最后,通过在 X 方程式中插入 6 来检查您的答复。

$\begin{aligned} \begin{array}{rcl} x + 12 & = & 18 \\ 6 + 12 & = & 18 \\ 18 & = & 18 \end{array} \end{aligned}$

The answer checks out.

::答案检查出来。

Review
::回顾

Solve each single-variable addition equation. Write your answer in the form:
::解析每个单变量附加方程式。请以窗体形式写入您的答复 :

$\begin{aligned} Variable = \underline{} . \end{aligned}$
::变数=_。

$\begin{aligned} x + 4 = 11 \end{aligned}$
::x+ 4= 11x+ 4= 11

$\begin{aligned} x + 11 = 22 \end{aligned}$
::x+11=22x+11=22

$\begin{aligned} x + 3 = 8 \end{aligned}$
::x+3=8x+3=8

$\begin{aligned} x + 12 = 20 \end{aligned}$
::x+12=20x+12=20

$\begin{aligned} x + 9 = 11 \end{aligned}$
::x + 9 = 11

$\begin{aligned} x + 8 = 30 \end{aligned}$
::x + 8 = 30

$\begin{aligned} 22 + x = 29 \end{aligned}$
::22+x=29

$\begin{aligned} 18 + x = 25 \end{aligned}$
::18+x=25

$\begin{aligned} 15 + x = 20 \end{aligned}$
::15+x=20

$\begin{aligned} 13 + x = 24 \end{aligned}$
::13+x=24

$\begin{aligned} x + 18 = 24 \end{aligned}$
::x + 18 = 24

$\begin{aligned} 23 + x = 33 \end{aligned}$
::23+x=33

$\begin{aligned} y + 18 = 31 \end{aligned}$
::y + 18 = 31

$\begin{aligned} 21 + x = 54 \end{aligned}$
::21+x=54

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。

Resources
::资源

章节大纲

常规

单变量加法方程

Solving Single-Variable Addition Equations ::解决单变式添加等同

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)

Resources ::资源

Solving Single-Variable Addition Equations
::解决单变式添加等同

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)

Resources
::资源