Course: 7.9 单一变量乘数等数

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单个变量乘数等数
Mr. Ricky’s Biology class is going to the botanical gardens to study the different kinds of flowers and plants that are there. The students have raised 68 dollars for tickets so far. The student rate for each ticket is 5 dollars. Mr. Ricky asked the students to figure out how many tickets they can buy. How many tickets can they afford with 68 dollars?
::Ricky先生的生物学课将前往植物园研究各种花卉和植物。到目前为止,学生们已经为票价筹集了68美元。每张票的学生费率是5美元。Ricky先生要求学生们弄清楚他们能买到多少票。他们能买到多少票?他们能买到多少票?他们买得起68美元?

In this concept, you will learn to solve equations with one variable and involve multiplication .
::在此概念中,您将学会用一个变量解析方程式,并涉及乘法。

Solving Single Variable Multiplication Equations
::解决单变量乘数等数

In the algebraic equation below, the variable $\begin{aligned} z \end{aligned}$ represents a number.
::在下面的代数方程式中,变量 z 代表一个数字。

$\begin{aligned} z \times 2 = 8 \end{aligned}$

::zx2=8

What number does $\begin{aligned} z \end{aligned}$ represent?
::z代表的号码是多少?

You can find out by asking yourself, “What number, when multiplied by 2, equals 8?”
::你可以问自己,“乘以2乘以2等于8时,多少数字等于8?”

Since $\begin{aligned} 4 \times 2 = 8 \end{aligned}$ , $\begin{aligned} z \end{aligned}$ must be equal to 4.
::由于 4x2=8, z必须等于 4 。

Sometimes, particularly with larger numbers, it can be more difficult to just 'know' the answer. Consider $\begin{aligned} z \times 7 = 105 \end{aligned}$ , since you may not know right away what number, multiplied by 7, equals 105, you should use another strategy for solving the equation .
::有时,特别是如果数字较大, 仅仅“ 知道” 答案可能更加困难。考虑 zx7=105, 因为您可能不知道数字是多少, 乘以7 等于 105, 您应该使用另一种策略来解决方程。

To solve a more challenging equation in which a variable is multiplied by a number, you can use the inverse operation of multiplication- division . To isolate the variable , divide both sides of the equation by the number the variable is multiplied by.
::要解决一个更具挑战性的方程式,其中变量乘以一个数字,您可以使用乘法差异的反向操作。要孤立变量,将方程式的两边除以变量乘以数字。

You can divide both sides of the equation by the same number and not change the equality because of the Division Property of Equality , which says that two things that are the same will still be the same if they are both divided by an equal amount. In math language, this looks like:
::你可以将等式的两边除以相同数字,而不会改变平等,因为“平等财产分配”规定,如果两者均以相同数额分开,两样东西将保持不变。在数学语言中,这看起来是:

If $\begin{aligned} a = b \end{aligned}$ and $\begin{aligned} c \neq 0 \end{aligned}$ , then $\begin{aligned} \frac{a}{c} = \frac{b}{c} \end{aligned}$ .
::如果a=b和c=c=0,那么ac=bc。

In other words, if you divide one side of an equation by a nonzero number, $\begin{aligned} c \end{aligned}$ , you must divide the other side of the equation by that same number, $\begin{aligned} c \end{aligned}$ , to keep the values on both sides equal.
::换句话说,如果将方程式的一面除以非零数, c, 则必须将方程式的另一面除以同一数字, c, 以使双方的数值相等。

Let’s look at an example.
::让我们举个例子。

Solve for $\begin{aligned} z \end{aligned}$ .
::解决z。

$\begin{aligned} z \times 7 = 105 \end{aligned}$

::zx7=105

In this equation, $\begin{aligned} z \end{aligned}$ is multiplied by 7. So, to isolate the variable $\begin{aligned} z \end{aligned}$ , you can divide both sides of the equation by 7.
::在这个方程中,z乘以7。所以,为了分离变量z, 你可以将方程的两边除以7。

First, using the division property of equality, divide both sides of the equation by 7.
::首先,利用平等分割财产,将等式两侧除以7。

$\begin{aligned} \begin{array}{rcl} z \times 7 & = & 105 \\ \frac{z \times 7}{7} & = & \frac{105}{7} \end{array} \end{aligned}$

::zx7=105zx77=1057

Next, separate the fraction $\begin{aligned} \frac{z \times 7}{7} \end{aligned}$ , and simplify.
::接下来,分离分数 zx77,并简化。

$\begin{aligned} \begin{array}{rcl} z \times \frac{7}{7} & = & 15 \\ z \times 1 & = & 15 \\ z & = & 15 \end{array} \end{aligned}$

::zx77=15zx1=15z=15

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

Here is another example.
::下面是另一个例子。

Solve for $\begin{aligned} r \end{aligned}$ : $\begin{aligned} - 8 r = 128 \end{aligned}$
::r: - 8r=128

In this equation, -8 is multiplied by $\begin{aligned} r \end{aligned}$ . So, using the division property of equality, you can divide both sides of the equation by -8 to solve for $\begin{aligned} r \end{aligned}$ .
::在此方程中, - 8 乘以 r。所以, 使用平等分割属性, 您可以将公式的两边除以 - 8 来解析 r 。

First, divide both sides of the equation by -8.
::首先,将等式的两边除以-8。

$\begin{aligned} \begin{array}{rcl} - 8 r & = & 128 \\ \frac{- 8 r}{- 8} & = & \frac{128}{- 8} \end{array} \end{aligned}$

::-8r=128-8r-8=128-8r-8=128-8

Next, separate the fraction $\begin{aligned} \frac{- 8 r}{- 8} \end{aligned}$ , and simplify.
::接下来,将分数-8-8分开,并简化。

$\begin{aligned} \begin{array}{rcl} \frac{- 8 r}{- 8} & = & \frac{128}{- 8} \\ 1 r & = & - 16 \\ r & = & - 16 \end{array} \end{aligned}$

::-8-8=128-81-81r}161616

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

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Mr. Ricky’s Biology class.
::先前,你曾因Ricky先生的生物学课而感到疑惑。

The students have raised 68 dollars for their trip and are wondering how many 5 dollar tickets they can buy.
::学生们为他们的行程募集了68美元,

First, write an equation to represent this information. Let $\begin{aligned} d \end{aligned}$ , be the number of tickets they can buy. You can say that $\begin{aligned} d \end{aligned}$ times the price of each ticket, 5 dollars, is 68 dollars.
::首先, 写一个方程式来代表这些信息。以他们能买到的票数来表示。您可以说, 每张票的价格是5美元, 乘以每张票的价格是68美元。

$\begin{aligned} 5 d = 68 \end{aligned}$

::5d=68

Next, use the division property of equality and divide both sides of the equation by 5.
::其次,使用平等分割财产,将等式两侧除以5。

$\begin{aligned} \frac{5 d}{5} = \frac{68}{5} \end{aligned}$

::5d5=685

Then, separate the fraction $\begin{aligned} \frac{5 d}{5} \end{aligned}$ and simplify.
::然后将分数5d5分离并简化

$\begin{aligned} \begin{array}{rcl} \frac{5}{5} d & = & 13.6 \\ 1 d & = & 13.6 \\ d & = & 13.6 \end{array} \end{aligned}$

::55d=13.61d=13.6d=13.6d=13.6

Next, interpret the result.
::接下来,解释结果。

The students can buy 13.6 tickets. You can’t buy .6 of a ticket. So, the students can only buy 13 tickets with 3 dollars left over.
::学生们可以买13.6张票。你买不到6张票。因此,学生们只能买13张票,剩下的3美元只能买到13张票。

The answer is the students can buy 13 tickets.
::答案是学生可以买13张票

Example 2
::例2

Sarvenaz earns $8 for each hour she works. She earned a total of $168 last week.
::Sarvenaz每工作一小时挣8美元,上周总共挣了168美元。

Write an equation to represent $\begin{aligned} h \end{aligned}$ , the number of hours she worked last week.
::写一个方程式来代表h, 也就是她上周工作的时间。

Determine how many hours Sarvenaz worked last week.
::确定Sarvenaz上周工作了多少小时

First, complete part a.
::第一,完整的第一部分。

Let $\begin{aligned} h \end{aligned}$ be the number of hours Sarvenaz worked. She earns $8 for each hour she works, so you multiply the number of hours she worked by $8 to find the total amount she earned. Write a multiplication equation.
::以Sarvenaz的工作小时数为例。她每工作小时挣8美元,所以你将她的工作小时数乘以8美元,以找到她所得的总金额。写一个乘法方程式。

$\begin{aligned} 8 h = 168 \end{aligned}$

::8h=168

Next, work on part b.
::接下来,关于B部分的工作。

Solve the equation $\begin{aligned} 8 h = 168 \end{aligned}$ to find $\begin{aligned} h \end{aligned}$ , the number of hours she worked last week.
::解决了8小时=168的方程找到她上周工作的时间

First, use the division property of equality to divide both sides of the equations by 8.
::首先,利用平等分割财产来将等式的两边除以8。

$\begin{aligned} \begin{array}{rcl} 8 h & = & 168 \\ \frac{8 h}{8} & = & \frac{168}{8} \end{array} \end{aligned}$

::8h=16888888=1688

Next, separate the fraction $\begin{aligned} \frac{8 h}{8} \end{aligned}$ and simplify.
::接下来,将第8h8项分离并简化。

$\begin{aligned} \begin{array}{rcl} \frac{8}{8} h & = & 21 \\ 1 h & = & 21 \\ h & = & 21 \end{array} \end{aligned}$

::88h=211h=21h=21

The answer is Sarvenaz works 21 hours last week.
::答案是Sarvenaz上周工作21小时

Solve each equation.
::解决每个方程式

Example 3
::例3

$\begin{aligned} - 4 x = 12 \end{aligned}$

::- 4x=12

First, use the division property of equality, and divide both sides of the equation by -4.
::首先,使用平等财产分割法,将等式的两侧除以-4。

$\begin{aligned} \begin{array}{rcl} - 4 x & = & 12 \\ \frac{- 4 x}{- 4} & = & \frac{12}{- 4} \end{array} \end{aligned}$

::-4x=12-4x-4=12-4=12-4

Next, separate the fraction $\begin{aligned} \frac{- 4 x}{- 4} \end{aligned}$ and simplify.
::下一步,将分数-4x-4分开并简化。

$\begin{aligned} \begin{array}{rcl} \frac{- 4}{- 4} x & = & - 3 \\ 1 x & = & - 3 \\ x & = & - 3 \end{array} \end{aligned}$

::-4 -4x31x3x3 -3

The answer is $\begin{aligned} x = - 3 \end{aligned}$ .
::答案是 x% 3 。

Example 4
::例4

$\begin{aligned} 8 a = 64 \end{aligned}$

::8a=64

First, use the division property of equality and divide both sides of the equation by 8.
::首先,使用平等分割财产,将等式的两边除以8。

$\begin{aligned} \begin{array}{rcl} 8 a & = & 64 \\ \frac{8 a}{8} & = & \frac{64}{8} \end{array} \end{aligned}$

::8a=648a8=648

Next, separate the fraction $\begin{aligned} \frac{8 a}{8} \end{aligned}$ and simplify.
::接下来,将第8a8项分开,并简化。

$\begin{aligned} \begin{array}{rcl} \frac{8}{8} a & = & 8 \\ 1 a & = & 8 \\ a & = & 8 \end{array} \end{aligned}$

::88a=81a=8a=8a=8

The answer is $\begin{aligned} a = 8 \end{aligned}$ .
::答案是a=8。

Example 5
::例5

$\begin{aligned} 9 b = 81 \end{aligned}$

::9b=81

First, use the division property of equality and divide both sides of the equation by 9.
::第一,使用平等分割财产,将等式两侧除以9。

$\begin{aligned} \begin{array}{rcl} 9 b & = & 81 \\ \frac{9 b}{9} & = & \frac{81}{9} \end{array} \end{aligned}$

::9b=819b9=819

Next, separate the fraction $\begin{aligned} \frac{9 b}{9} \end{aligned}$ and simplify.
::接下来,将第9b9项分离并简化。

$\begin{aligned} \begin{array}{rcl} \frac{9}{9} b & = & 9 \\ 1 b & = & 9 \\ b & = & 9 \end{array} \end{aligned}$

::99b=91b=9b=9

The answer is $\begin{aligned} b = 9 \end{aligned}$ .
::答案是b=9。

Review
::回顾

Solve each single variable multiplication equation for the missing value.
::为缺失值解决每个单个变量乘法方程式。

$\begin{aligned} 4 x = 16 \end{aligned}$
::4x=16

$\begin{aligned} 6 x = 72 \end{aligned}$
::6x=72

$\begin{aligned} - 6 x = 72 \end{aligned}$
::-6x=72

$\begin{aligned} - 3 y = 24 \end{aligned}$
::-3y=24

$\begin{aligned} - 3 y = - 24 \end{aligned}$
::- 3 - 24岁

$\begin{aligned} - 5 x = - 45 \end{aligned}$
::- 5x45

$\begin{aligned} - 1.4 x = 2.8 \end{aligned}$
::-1.4x=2.8

$\begin{aligned} 3.5 a = 7 \end{aligned}$
::3.5a=7

$\begin{aligned} 7 a = - 49 \end{aligned}$
::7a49

$\begin{aligned} 14 b = - 42 \end{aligned}$
::14b42

$\begin{aligned} 24 b = - 48 \end{aligned}$
::24b48

$\begin{aligned} - 24 b = - 48 \end{aligned}$
::-24b48

$\begin{aligned} 34 b = - 102 \end{aligned}$
::34b102

$\begin{aligned} 84 x = 252 \end{aligned}$
::84x=252

$\begin{aligned} - 84 x = - 252 \end{aligned}$
::-84x 252

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

Section outline

General

单个变量 乘数等数

Solving Single Variable Multiplication Equations ::解决单变量乘数等数

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

单个变量乘数等数

Solving Single Variable Multiplication Equations
::解决单变量乘数等数

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