课程： 12.6单变量减法方程

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单变量减法方程
Steve and his family are going on a mini-vacation or a weekend getaway to the beach. Steve and his two sisters are excited about staying at a hotel on the beach. When they arrive at the hotel, Steve’s father, Mr. Richards, goes to the registration desk to check for his reservation. The manager tells him it will cost $288 after he receives a 10% discount on the two-room family suite. Mr. Richards thanks the manager for the discount. How can Mr. Richard figure out how much the suite was before the discount?
::史蒂夫和他的家人正在去一个迷你空地或周末离开海滩。史蒂夫和他的两个姐妹对住在海滩上的一家旅馆很兴奋。当他们到达旅馆时,史蒂夫的父亲理查德先生去登记台查看他的预订。经理告诉他,在两间房间的家庭套房得到10%的折扣后,费用将达288美元。理查德先生感谢经理的折扣。理查德先生怎么知道折扣之前套房是多少?

In this concept, you will learn to solve single- variable subtraction equations.
::在此概念中,您将学会解析单变量减法方程式。

Solving Single-Variable Subtraction Equations
::解决单变式减法等量

You can use inverses to solve single-variable subtraction equations.
::您可以使用逆数解析单变量减法方程式。

The goal is to get the variable alone on one side of the equal signs and using the inverse of subtraction – addition , can help you do this. Keep in mind that what you do to one side of the equation you have to do to the other side.
::目标是将变量单放在等号的一面,使用反减法 — — 添加,可以帮助你做到这一点。记住,你对等方程的一面所做的一切,对另一面必须做的。

Here is an equation.
::这里有一个方程式。

$\begin{aligned} x - 12 = 40 \end{aligned}$

To solve this equation, you can use the inverse of subtraction (addition) and add 12 to both sides of the equation. That will help to get the variable alone and solve the problem.
::要解决此等式, 您可以使用减法( 附加) 的反向, 并在等式的两侧添加 12 。这样可以帮助将变量单独定位并解决问题。

First, identify the number being subtracted from the variable.
::首先,标明从变量中减去的数字。

-12

Next, using the inverse of 12, add 12 to both sides of the equation.
::接下来,用12的反数, 方程的两侧加12。

$\begin{aligned} \begin{array}{rcl} x - 12 = 40 \\ \underline{+ 12 + 12} \\ x - 0 = 52 \end{array} \end{aligned}$

The plus 12 cancels the minus 12 on the left side of the equation, leaving only $\begin{aligned} x \end{aligned}$ . On the other side of the equation, 12 and 40 are added to get 52.
::加12取消方程左侧的减12,只留下x。在方程的另一侧,加上12和40,得出52。

The answer is $\begin{aligned} x = 52 \end{aligned}$ .
::答案是 x = 52 。

To check this answer, substitute it back into the original problem and see if the statement is true.
::要检查此答案, 请将其换回原来的问题, 看看声明是否真实。

$\begin{aligned} \begin{array}{rcl} x - 12 & = & 40 \\ 52 - 12 & = & 40 \\ 40 & = & 40 \end{array} \end{aligned}$

The answer is true.
::答案是真实的。

Sometimes, in single-variable subtraction equations, the variable will be subtracted from a number. Take a look at the equation below.
::有时,在单变量减法方程式中,变量将从数字中减去。看看下面的方程式。

$\begin{aligned} 32 - x = 7 \end{aligned}$

One way to solve this equation without getting into positive and negative numbers is to simply turn it into a single-variable addition equation by using the inverse of subtraction and adding $\begin{aligned} x \end{aligned}$ to both sides.
::解决这个等式而不进入正数和负数的一个办法是,简单地通过使用逆减法和在两侧增加x,将它变成一个单一的可变加方程。

First, since the variable $\begin{aligned} x \end{aligned}$ is being subtracted, add $\begin{aligned} x \end{aligned}$ to both sides.
::首先,由于变量 x 被减去,在两侧添加 x 。

$\begin{aligned} \begin{array}{rcl} 32 - x = 7 \\ \underline{+ x = 7 + x} \\ 32 = 7 + x \end{array} \end{aligned}$

Now, you have a single-variable addition equation.
::现在,你有一个单变量附加方程式。

$\begin{aligned} x + 7 = 32 \end{aligned}$

Next, solve the addition equation by using the inverse of addition and subtracting 7 from both sides.
::其次,通过使用反增量和从双方减去7,解决增加方程问题。

$\begin{aligned} \begin{array}{rcl} x + 7 = 32 \\ \underline{- 7 = - 7} \\ x = 25 \end{array} \end{aligned}$

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

Then, check your answer by substituting 25 for $\begin{aligned} x \end{aligned}$ in the original equation.
::然后,检查你的答案,在原来的方程中,以25取代xx。

$\begin{aligned} \begin{array}{rcl} 32 - x & = & 7 \\ 32 - 25 & = & 7 \\ 7 & = & 7 \end{array} \end{aligned}$

The answer checks out.
::答案检查出来。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Steve and his family’s weekend beach getaway.
::更早之前, 史蒂夫和他的家人在周末海滩外出时遇到麻烦。

Mr. Richards wants to know the usual cost of the room if he paid $288 after a $32 discount.
::Richards先生想知道如果在扣除32美元折扣后支付288美元,房间的通常费用是多少。

First, write an equation for the situation.
::首先,写一个情况方程式

The total cost ( $\begin{aligned} x \end{aligned}$ ) minus 32 (discount) is 288.
::总费用(x)减32(贴现)为288。

Or
::或

$\begin{aligned} x - 32 = 288 \end{aligned}$

Next, identify the number being subtracted from the variable.
::下一步,请标明从变量中减去的数值。

- 32

Then, using the inverse of -32, add 32 to both sides of the equation.
::然后,用 -32的逆数,在方程的两侧加32。

$\begin{aligned} \begin{array}{rcl} x - 32 = 288 \\ \underline{+ 32 + 32} \\ x - 0 = 320 \end{array} \end{aligned}$

The answer is $\begin{aligned} x = $ 320 \end{aligned}$ .
::答案是x=320美元。

To check this answer, substitute $320 back into the original problem and see if the statement is true.
::要检查这个答案, 替换 320 美元回到原来的问题, 看看声明是否真实。

$\begin{aligned} \begin{array}{rcl} x - 32 & = & 288 \\ 320 - 32 & = & 288 \\ 320 & = & 320 \end{array} \end{aligned}$

The answer is true.
::答案是真实的。

The room cost $320 before the discount.
::房价在折扣前要320美元

Example 2
::例2

Solve the following equation.
::解决以下方程式。

$\begin{aligned} y - 21 = 59 \end{aligned}$

First, identify the number being subtracted from the variable.
::首先,标明从变量中减去的数字。

-21

Next, using the inverse of 21, add 21 to both sides of the equation.
::接下来,用倒数21,在方程的两侧加21。

$\begin{aligned} \begin{array}{rcl} y - 21 = 59 \\ \underline{+ 21 + 21} \\ y - 0 = 80 \end{array} \end{aligned}$

The answer is $\begin{aligned} y = 80 \end{aligned}$ .
::答案是y=80。

To check this answer, substitute it back into the original problem and see if the statement is true.
::要检查此答案, 请将其换回原来的问题, 看看声明是否真实。

$\begin{aligned} \begin{array}{rcl} y - 21 & = & 59 \\ 80 - 21 & = & 59 \\ 59 & = & 59 \end{array} \end{aligned}$

The answer is true.
::答案是真实的。

Solve each equation and write your answer in the format: $\begin{aligned} x = \underline{} \end{aligned}$ .
::解析每个方程式并用格式写出答案: x = _。

Example 3
::例3

$\begin{aligned} x - 9 = 22 \end{aligned}$

First, identify the number being subtracted from the variable.
::首先,标明从变量中减去的数字。

-9

Next, using the inverse of 9, add 9 to both sides of the equation.
::接下来,用倒数9,在方程的两侧加9。

$\begin{aligned} \begin{array}{rcl} x - 9 = 22 \\ \underline{+ 9 + 9} \\ x - 0 = 31 \end{array} \end{aligned}$

The answer is $\begin{aligned} x = 31 \end{aligned}$ .
::答案是 x = 31 。

To check this answer, substitute it back into the original problem and see if the statement is true.
::要检查此答案, 请将其换回原来的问题, 看看声明是否真实。

$\begin{aligned} \begin{array}{rcl} x - 9 & = & 22 \\ 31 - 9 & = & 22 \\ 22 & = & 22 \end{array} \end{aligned}$

The answer is true.
::答案是真实的。

Example 4
::例4

$\begin{aligned} x - 3 = 46 \end{aligned}$

First, identify the number being subtracted from the variable.
::首先,标明从变量中减去的数字。

-3

Next, using the inverse of -3, add 3 to both sides of the equation.
::接下来,使用 - 3的逆数, 在方程的两侧增加 3 。

$\begin{aligned} \begin{array}{rcl} x - 3 = 46 \\ \underline{+ 3 + 3} \\ x - 0 = 49 \end{array} \end{aligned}$

The answer is $\begin{aligned} x = 49 \end{aligned}$ .
::答案是 x = 49 。

To check this answer, substitute it back into the original problem and see if the statement is true.
::要检查此答案, 请将其换回原来的问题, 看看声明是否真实。

$\begin{aligned} \begin{array}{rcl} x - 3 & = & 46 \\ 49 - 3 & = & 46 \\ 46 & = & 46 \end{array} \end{aligned}$

The answer is true.
::答案是真实的。

Example 5
::例5

$\begin{aligned} x - 7 = 23 \end{aligned}$

First, identify the number being subtracted from the variable.
::首先,标明从变量中减去的数字。

-7

Next, using the inverse of -7, add 7 to both sides of the equation.
::接下来,使用 - 7 的逆数,在方程的两侧增加 7。

$\begin{aligned} \begin{array}{rcl} x - 7 = 23 \\ \underline{+ 7 + 7} \\ x - 0 = 30 \end{array} \end{aligned}$

The answer is $\begin{aligned} x = 30 \end{aligned}$ .
::答案是 x = 30 。

To check this answer, substitute it back into the original problem and see if the statement is true.
::要检查此答案, 请将其换回原来的问题, 看看声明是否真实。

$\begin{aligned} \begin{array}{rcl} x - 7 & = & 23 \\ 30 - 7 & = & 23 \\ 23 & = & 23 \end{array} \end{aligned}$

The answer is true.

::答案是真实的。

Review
::回顾

Solve each single-variable subtraction problem using the inverse operation. Write your answer in the form: $\begin{aligned} variable = \underline{} \end{aligned}$ .
::使用反向操作解决每个单变量减法问题。请以窗体写您的回答: 变量 = _ 。

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

$\begin{aligned} x - 7 = 17 \end{aligned}$
::x - 7 = 17

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

$\begin{aligned} z - 6 = 22 \end{aligned}$
::zz- 6 = 22

$\begin{aligned} y - 9 = 11 \end{aligned}$
::y - 9 = 11

$\begin{aligned} b - 5 = 12 \end{aligned}$
::b - 5 = 12

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

$\begin{aligned} y - 7 = 2 \end{aligned}$
::y - 7 = 2

$\begin{aligned} x - 9 = 1 \end{aligned}$
::x-9=1x-9=1

$\begin{aligned} x - 19 = 15 \end{aligned}$
::x - 19 = 15

$\begin{aligned} x - 18 = 12 \end{aligned}$
::x - 18 = 12

$\begin{aligned} x - 29 = 31 \end{aligned}$
::x-29 = 31

$\begin{aligned} x - 15 = 62 \end{aligned}$
::x - 15 = 62

$\begin{aligned} x - 22 = 45 \end{aligned}$
::x - 22 = 45

$\begin{aligned} x - 19 = 37 \end{aligned}$
::x - 19 = 37

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

单变量减法方程

Solving Single-Variable Subtraction Equations ::解决单变式减法等量

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)

Resources ::资源

Solving Single-Variable Subtraction Equations
::解决单变式减法等量

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)

Resources
::资源