节: 使用精神数学和替代方法解决和检查单变等同 | 1.11 使用精神数学和替代方法解决和检查单变等同

章节大纲

Maria and her mother are shopping for a dress for Maria’s school dance. The dress Maria likes the most is $120. She asks her mom, “If I give you my cleaning money for the next six weeks, will you pay the other half?”
::玛利亚和她的母亲正在为玛利亚的学校舞蹈买裙子。玛利亚最喜欢的裙子是120美元。她问她妈妈 : “ 如果我把接下来六个星期的清洁费给你,你会付另一半吗? ”

Maria is paid the same amount of money each week to clean her neighbor’s house. How can you use the information given here to determine how much money Maria earns each week?
::玛利亚每周得到同等数额的钱来打扫邻居家。您如何使用这里提供的信息来决定玛利亚每周挣多少钱?

In this concept, you will learn to solve and check single variable equations using mental math and substitution.
::在此概念中,您将学会使用心理数学和替代来解析和检查单变量方程式。

Solving Equations Using Mental Math
::使用心理数学解决等同

An equation is a statement of equality of two mathematical expressions. The quantity on one side of the equals sign must have the same value as the quantity on the other side of the equals sign. For example, $\begin{aligned} 11 - 6 = 5 \end{aligned}$ , $\begin{aligned} x + 4 = 12 \end{aligned}$ , $\begin{aligned} 3 x - 6 = 9 \end{aligned}$ , and $\begin{aligned} \frac{x}{2} - 5 = 4 \end{aligned}$ are all equations with a single variable. Remember, a variable is a letter that represents a quantity.
::等式是两个数学表达式相等的语句。等值符号一面的数量必须与等值符号另一面的数量具有相同值。例如, 11- 6=5, x+4=12, 3x- 6=9, 和 x2- 5=4 是一个变量的方程式。记住, 一个变量是代表一个数量的字母。

When working with an equation involving a single variable, you are looking for the number that gives a true statement when you replace the variable by the number. This process is referred to as solving the equation. The number that gives the true statement is said to “satisfy” the equation and is called the solution or root of the equation.
::当使用涉及单一变量的方程式工作时,您正在寻找一个数字,该数字在用数字替换变量时给出真实的语句。这个过程被称为解析方程式。给出真实语句的数字被称为“满足”方程式,被称为方程式的解决方案或根。

Let’s look at an example of solving an equation with a single variable.
::让我们看看用单一变量解决方程式的例子。

Solve the equation $\begin{aligned} x + 4 = 12 \end{aligned}$ .
::解析方程式 x+4=12 。

You can solve this equation using mental math.
::你可以用心理数学解决这个方程式

Ask yourself, “What number added to 4 gives 12 or what is the result of subtracting 4 from 12?” Either way, the solution is 8 because $\begin{aligned} 8 + 4 = 12 \end{aligned}$ and $\begin{aligned} 12 - 4 = 8 \end{aligned}$ . You have solved the equation and can express your solution as $\begin{aligned} x = 8 \end{aligned}$ .
::问问你自己,“4号中增加的多少数字给12分12分,还是从12分中减去4分的结果是什么?” 无论哪种方式,解决办法是8分,因为8+4=12和12-4=8分。你已经解决了方程,可以以x=8表示你的解决方案。

To check your answer, return to the equation you were given to solve and substitute $\begin{aligned} x = 8 \end{aligned}$ into the equation.
::要检查您的答案, 请返回到您被赋予的方程式中解析和替换 x=8 的方程式。

$\begin{aligned} \begin{array}{rcl} x + 4 & = & 12 \\ 8 + 4 & = & 12 \end{array} \end{aligned}$

::x+4=128+4=12

Then, add the numbers on the left side of the equals sign.
::然后,在等同的牌子左侧加上数字。

$\begin{aligned} \begin{array}{rcl} 8 + 4 & = & 12 \\ 12 & = & 12 \end{array} \end{aligned}$

The solution of $\begin{aligned} x = 8 \end{aligned}$ gives a true statement.
::x=8 的解决方案给出了真实的语句。

Let’s look at one more example.
::让我们再举一个例子。

Solve the equation $\begin{aligned} 3 x = 18 \end{aligned}$ .
::解决方程式 3x=18

You can figure out this solution by using mental math. Ask yourself, “What number times 3 gives 18 or how many times 3 divides into 18?” Either way, the solution is 6 because $\begin{aligned} 3 \times 6 = 18 \end{aligned}$ and $\begin{aligned} 18 \div 3 = 6 \end{aligned}$ .
::您可以使用心理数学来找到这个解决方案。问问你自己 , “ 多少乘以 3 等于 18 或乘以多少乘以 3 等于 18 = 18 ? ” 无论如何, 答案是 6, 因为 3x6= 18 和 18= 3 = 6 。

The solution is $\begin{aligned} x = 6 \end{aligned}$ .
::溶液为 x=6 。

To check your answer, return to the equation you were given to solve and substitute $\begin{aligned} x = 6 \end{aligned}$ into the equation.
::要检查您的答案, 请返回到您被赋予的方程式中解析和替换 x=6 的方程式。

$\begin{aligned} \begin{array}{rcl} 3 x & = & 18 \\ 3 (6) & = & 18 \end{array} \end{aligned}$

::3x=183(6)=18

Now multiply the numbers on the left side of the equals sign.
::现在将等号左侧的数字乘以。

$\begin{aligned} 18 = 18 \end{aligned}$

The solution of $\begin{aligned} x = 6 \end{aligned}$ gives a true statement.
::x=6 的解决方案给出了真实的语句。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Maria and her new dress.
::之前,你对Maria和她的新裙子有个问题

You want to know how much money Maria earns each week, if six weeks worth of pay is equal to half the cost of the $120 dress.
::你想知道Maria每周挣多少钱? 如果6周的薪水相当于120美元礼服的一半。

First, divide $\begin{aligned} $ 120.00 \div 2 \end{aligned}$ to figure out how much Maria and her mom are each paying for the gown.
::首先,分出120美元来弄清楚 Maria和她妈妈每人要付多少钱来买礼服

$\begin{aligned} \frac{120}{2} = 60 \end{aligned}$

Next, write an equation to show that Maria is paying 6 times some amount of money $\begin{aligned} (x) \end{aligned}$ equal to $60.
::接下来,写一个方程式显示玛丽亚支付6倍一些金额(x)等于60美元。

$\begin{aligned} 6 x = 60 \end{aligned}$

::6x=60

Then, ask yourself, “Six times what number equals 60?”
::那么,问问你自己,“数字的六倍等于60?”

The answer is 10.
::答案是10个

The solution is $\begin{aligned} x = 10 \end{aligned}$ dollars.
::溶液是x=10美元。

Example 2
::例2

Solve the equation $\begin{aligned} 3 x - 6 = 9 \end{aligned}$ .
::解析方程式 3x - 6=9 。

The equation is asking you to find 3 times what number $\begin{aligned} (x) \end{aligned}$ take away 6 equals nine.
::方程式要求您找到3倍于数字( x) 的乘以 6 等于 9 。

You can use mental math to solve this equation.
::你可以用心理数学来解析这个方程式

First, ask yourself, “What number subtract 6 equals 9?”
::首先,问问你自己,“减去6等于9是多少?”

The answer is 15.
::答案是15岁

Next, ask yourself, “Three times what number $\begin{aligned} (x) \end{aligned}$ equals 15?”
::接下来,问自己,“数字(x)的三倍等于15?”

The answer is 5.
::答案是5。

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

To check your answer, return to the equation you were given to solve and substitute $\begin{aligned} x = 5 \end{aligned}$ into the equation.
::要检查您的答案, 请返回到您被赋予的方程式中解析和替换 x=5 的方程式。

$\begin{aligned} \begin{array}{rcl} 3 x - 6 & = & 9 \\ 3 (5) - 6 & = & 9 \end{array} \end{aligned}$

::3x-6=93(5)-6=9

First, multiply $\begin{aligned} 3 (5) = 15 \end{aligned}$ on the left side of the equals sign.
::首先,乘以3(5)=15,在等号的左侧。

$\begin{aligned} 15 - 6 = 9 \end{aligned}$

Next, subtract the numbers $\begin{aligned} 15 - 6 = 9 \end{aligned}$ on the left side of the equals sign.
::下一步,减去等号左侧15-6=9的数字。

$\begin{aligned} 9 = 9 \end{aligned}$

The root is $\begin{aligned} x = 5 \end{aligned}$ .
::根为 x=5 。

Example 3
::例3

Solve and check the equation $\begin{aligned} \frac{x}{2} = 12 \end{aligned}$ using mental math.
::使用心理数学解决并检查方程式 x2=12 。

First, determine what the equation is asking you to find.
::首先,决定方程式要你找到什么

The equation is asking you to find what number $\begin{aligned} (x) \end{aligned}$ divided by two equals 12.
::方程式要求您找到数字( x) 除以 2 等于 12 。

The number is 24 because $\begin{aligned} 24 \div 2 = 12 \end{aligned}$ .
::号码是24 因为242=12

The root is $\begin{aligned} x = 24 \end{aligned}$ .
::根是 x=24 。

CHECK:
::切克:

First, substitute $\begin{aligned} x = 24 \end{aligned}$ into the equation.
::首先,在方程中替换 x=24。

$\begin{aligned} \frac{24}{2} = 12 \end{aligned}$

Then, divide 24 by 2 on the left side of the equals sign.
::然后,在等同的牌子左侧,将24除以2。

$\begin{aligned} \begin{array}{rcl} \overset{12}{2 \bar{) 24}} \\ \underline{- 2} \\ 04 \\ \underline{- 4} \\ 0 \\ 12 = 12 \end{array} \end{aligned}$

Example 4
::例4

Determine whether $\begin{aligned} x = 6 \end{aligned}$ is the solution to the following equation:
::确定 x=6 是否是以下方程的解决方案:

$\begin{aligned} 3 x - 8 = 12 \end{aligned}$

::3x-8=12

First, substitute $\begin{aligned} x = 6 \end{aligned}$ into the given equation.
::首先,在给定方程中替换 x=6 。

$\begin{aligned} 3 (6) - 8 = 12 \end{aligned}$

Next, multiply $\begin{aligned} 3 (6) = 18 \end{aligned}$ to clear the parenthesis.
::下一个乘以 3(6)=18来清除括号。

$\begin{aligned} 18 - 8 = 12 \end{aligned}$

Then, subtract the numbers on the left side of the equals sign. $\begin{aligned} 18 - 8 = 10 \end{aligned}$
::然后减去等号左侧的数字。 18- 8=10

$\begin{aligned} 10 = 12 \end{aligned}$

This is not a true statement.
::这不是一个真实的说法。

The answer is NOT $\begin{aligned} x = 6 \end{aligned}$ .
::答案不是 x=6 。

Review
::回顾

Solve each equation using mental math. Be sure to check each answer by substituting your solution back into the original problem. Then simplify to see if the equation expresses a true statement.
::使用心理数学来解析每个方程式。请确定要检查每个答案, 将您的解决方案替换回原来的问题。然后简化以查看方程式是否表达真实的语句。

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

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

$\begin{aligned} x - 19 = 40 \end{aligned}$
::x-19=40

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

$\begin{aligned} 4 x = 24 \end{aligned}$
::4x=24

$\begin{aligned} 6 x = 36 \end{aligned}$
::6x=36

$\begin{aligned} 9 x = 81 \end{aligned}$
::9x=81

$\begin{aligned} \frac{y}{5} = 2 \end{aligned}$
::y5=2 y5=2

$\begin{aligned} \frac{a}{8} = 5 \end{aligned}$
::a8=5

$\begin{aligned} \frac{12}{b} = 6 \end{aligned}$
::12b=6

$\begin{aligned} 6 x + 3 = 27 \end{aligned}$
::6x+3=27

$\begin{aligned} 8 y - 2 = 54 \end{aligned}$
::8-2=54

$\begin{aligned} 3 b + 12 = 30 \end{aligned}$
::3b+12=30

$\begin{aligned} 9 y - 7 = 65 \end{aligned}$
::9-7=65

$\begin{aligned} 12 a - 5 = 31 \end{aligned}$
::12a-5=31

$\begin{aligned} \frac{x}{2} + 4 = 8 \end{aligned}$
::x2+4=8

$\begin{aligned} \frac{x}{4} + 3 = 7 \end{aligned}$
::x4+3=7

$\begin{aligned} \frac{10}{x} + 9 = 14 \end{aligned}$
::10x+9=14 10x+9=14

$\begin{aligned} 5 a - 12 = 33 \end{aligned}$
::5a-12=33

$\begin{aligned} 7 b - 9 = 33 \end{aligned}$
::7b-9=33

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

Solving Equations Using Mental Math ::使用心理数学解决等同

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Solving Equations Using Mental Math
::使用心理数学解决等同

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源