课程： 12.9单变量除法方程

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单变量除法方程
It’s Teacher’s Planning Day at Jerry’s school, so Jerry and four of his friends have decided to spend the day at the local amusement park. Jerry and his friends rarely ride any of the attractions. They mostly spend their time playing video games in the game center. Each of the boys needs 15 tokens to play all of their favorite games. They nominate Jerry to stand in line at the token booth to purchase their tokens. As he waits in line, Jerry tries to figure out how many tokens he needs to get so he and his four friends will each have 15 tokens. How can Jerry use a single variable division equation to solve this problem?
::今天是Jerry学校的教师计划日,因此Jerry和他的四个朋友决定在当地游乐公园度过一天。 Jerry和他的朋友很少乘坐任何游乐场。他们大多花时间在游戏中心玩电子游戏。男孩们都需要15个纪念品来玩他们最喜欢的游戏。他们提名Jerry在代号亭排队购买他们的纪念品。当他排队时,Jerry试图弄清楚他需要多少纪念品才能拿到,因此他和他的四个朋友每人会有15个纪念品。杰瑞怎么能用一个可变的分部方程式来解决这个问题呢?

In this concept, you will learn how to solve single-variable division equations.
::在这个概念中,你将学会如何解决单一可变分裂方程式。

Solving Single-Variable Division Equations
::解决单一可变分区

Division equations require you to use multiplication to solve for the variable. Remember that multiplication is the inverse operation for division.
::分部方程式要求您使用乘法解析变量。记住乘法是分区的反向操作。

There are two different types of division equations that you will be solving. Let’s look at a problem that is the first type of division equation.
::您将要解决的分裂方程式有两种不同类型。让我们来研究一个第一种分裂方程式的问题。

$\begin{aligned} \frac{x}{3} = 12 \end{aligned}$

This type of division problem has a missing numerator. You don’t know the value of the numerator so you use a variable in place of the unknown number.
::这种分区问题类型有一个缺失的分子。您不知道分子的价值, 所以用变量代替未知的数字。

To figure out the numerator, multiply the denominator with the value on the right side of the equals.
::要找出分子, 将分母乘以等值右侧的值。

$\begin{aligned} \begin{array}{rcl} \frac{x}{3} & = & 12 \\ x & = & 3 (12) \\ x & = & 36 \end{array} \end{aligned}$

To check the answer, substitute it back into the original equation for the variable. If one side equals the other side, then your work is accurate and correct.
::要检查答案, 请将其替换为变量的原始方程。如果一方等于另一方, 那么您的工作是准确和正确的。

$\begin{aligned} \begin{array}{rcl} \frac{36}{3} & = & 12 \\ 12 & = & 12 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Now let’s look at the second type of division equation – one where the denominator is an unknown variable.
::现在让我们看看第二种分裂方程式 — — 分母是一个未知变量。

$\begin{aligned} \frac{4}{x} = 2 \end{aligned}$

To solve this equation you need to multiply the denominator with the value on the right side of the equals. In this case, the denominator is a variable. Multiply it by two and rewrite the problem.
::要解析此等式, 您需要将分母乘以等值右侧的值。在此情况下, 分母是一个变量。乘以 2 并重写问题。

$\begin{aligned} \begin{array}{rcl} \frac{4}{x} & = & 2 \\ 2 x & = & 4 \end{array} \end{aligned}$

Now you have a multiplication problem to solve. Solve it by using division. To get the variable alone, divide both sides of the equation by 2.
::现在您需要解决一个乘法问题。使用除法解决它。要单独获得变量, 将方程的两边除以 2 。

$\begin{aligned} \begin{array}{rcl} \frac{2 x}{2} & = & \frac{4}{2} \\ x & = & 2 \end{array} \end{aligned}$

To check your work, substitute the value of $\begin{aligned} x \end{aligned}$ back into the original problem.
::检查您的工作, 替换 x 的值返回到原始问题。

$\begin{aligned} \begin{array}{rcl} \frac{4}{x} & = & 2 \\ \frac{4}{2} & = & 2 \\ 2 & = & 2 \end{array} \end{aligned}$

It is accurate.
::这是准确的。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Jerry and his friends at the arcade.
::早些时候,有人给了你一个问题关于杰瑞和他的朋友在街机上。

Jerry needs to get enough tokens so that he and his four friends will each have 15 tokens. How can Jerry use a single variable division equation to figure out how many tokens he needs to purchase?
::Jerry需要足够的代币,以便他和他的四个朋友每人有15个代币。 Jerry怎么能用一个可变分部方程式来计算他需要购买多少代币呢?

First, mentally write an equation to represent the situation.
::首先,在精神上写一个方程式来代表情况。

$\begin{aligned} \frac{x}{5} = 15 \end{aligned}$

This equation means, an unknown number of tokens $\begin{aligned} (x) \end{aligned}$ divided 5 is 15.
::这个方程式表示, 数量不详的标记( x) 5 等于 15 。

Next, use the inverse of division, which is multiplication, and multiply the denominator (5) and the number (15) on the other side of the equal sign.
::其次,在等号的另一侧,使用正乘法的反差乘法,乘以分母(5)和数(15)。

$\begin{aligned} 5 \times 15 = 75 \end{aligned}$

That leaves $\begin{aligned} x \end{aligned}$ alone on one side of the equal signs.
::这样就让X单独出现在等同的标志的一边了

$\begin{aligned} x = 75 \end{aligned}$

Then, check the answer by substituting it back into the original equation for the variable. If one side equals the other side, then your work is accurate and correct.
::然后,检查答案,将其替换为变量的原始方程。如果一方等于另一方,那么你的工作是准确和正确的。

$\begin{aligned} \begin{array}{rcl} \frac{x}{5} & = & 15 \\ \frac{75}{5} & = & 15 \\ 15 & = & 15 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Solve the equations and write your answer in the form $\begin{aligned} Variable = \underline{} . \end{aligned}$
::解析方程式,并以变数 = _的形式写出答案。

Example 2
::例2

$\begin{aligned} \frac{x}{4} = 6 \end{aligned}$

First, use the inverse of division, which is multiplication, and multiply the denominator (4) and the number on the other side of the equal sign, 6.
::首先,使用正乘法的反差,乘法是乘法,乘法是分母(4)和等号6的另一侧的数字。

$\begin{aligned} 4 \times 6 = 24 \end{aligned}$

That leaves $\begin{aligned} x \end{aligned}$ alone on one side of the equal signs and 24 on the other side.
::这样就让X单独出现在等标的一面,24个在另一面。

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

Next, check the answer by substituting it back into the original equation for the variable. If one side equals the other side, then your work is accurate and correct.
::下一步, 请检查答案, 将其替换为变量的原始方程式。如果一方等于另一方, 那么您的工作是准确和正确的。

$\begin{aligned} \begin{array}{rcl} \frac{x}{4} & = & 6 \\ \frac{24}{4} & = & 6 \\ 6 & = & 6 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Example 3
::例3

$\begin{aligned} \frac{x}{5} = 7 \end{aligned}$

First, use the inverse of division, which is multiplication, and multiply the denominator (5) and 7.
::首先,使用分数的反向,即乘法,乘以分母(5)和7。

$\begin{aligned} 5 \times 7 = 35 \end{aligned}$

That leaves $\begin{aligned} x \end{aligned}$ alone on one side of the equal signs and 35 on the other side.
::这样就让X单独出现在等号的一面,35个在另一面。

$\begin{aligned} x = 35 \end{aligned}$

Next, check the answer by substituting it back into the original equation for the variable.
::下一步,检查答案,将其替换为变量的原始方程。

$\begin{aligned} \begin{array}{rcl} \frac{x}{5} & = & 7 \\ \frac{35}{5} & = & 7 \\ 7 & = & 7 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Example 4
::例4

$\begin{aligned} \frac{x}{2} = 3 \end{aligned}$

First, use the inverse of division, which is multiplication, and multiply the denominator (2) and the number (3) on the other side of the equal sign.
::首先,使用正乘法的反除法,乘法是乘法,乘法是分母(2)和数(3),在等号的另一侧。

$\begin{aligned} 2 \times 3 = 6 \end{aligned}$

That leaves $\begin{aligned} x \end{aligned}$ alone on one side of the equal signs and 6 on the other side.
::这让X单独出现在相同标志的一面,6在另一面。

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

Next, check the answer by substituting it back into the original equation for the variable. If one side equals the other side, then your work is accurate and correct.
::下一步, 请检查答案, 将其替换为变量的原始方程式。如果一方等于另一方, 那么您的工作是准确和正确的。

$\begin{aligned} \begin{array}{rcl} \frac{x}{2} & = & 3 \\ \frac{6}{2} & = & 3 \\ 3 & = & 3 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Example 5
::例5

$\begin{aligned} \frac{12}{x} = 6 \end{aligned}$

First, use the inverse of division, which is multiplication, and multiply the denominator $\begin{aligned} (x) \end{aligned}$ and the number (6) on the other side of the equal sign.
::首先,在等号的另一侧,使用乘法的反差,乘以分母(x)和数(6)。

$\begin{aligned} 6 (x) = 6 x \end{aligned}$

That leaves 12 alone on one side of the equal signs and $\begin{aligned} 6 x \end{aligned}$ on the other side.
::平等标志的一面只有12个,另一面6个。

$\begin{aligned} 12 = 6 x \end{aligned}$

Now you have a multiplication problem to solve. Solve it by using division.
::现在您需要解决一个乘法问题。使用除法解决它。

Next, to get the variable alone, divide both sides of the equation by 6.
::其次,为了单独获得变量,将方程的两边除以6。

$\begin{aligned} \frac{12}{6} = \frac{6 x}{6} \end{aligned}$

The 6’s cancel each other, leaving $\begin{aligned} x = 2 \end{aligned}$ .
::6人相互取消,留下x=2。

Then, substitute 2 for $\begin{aligned} x \end{aligned}$ in the original problem to check your work.
::然后用原来的问题中的 x 替换 2 替换 x 来检查您的工作。

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

It is accurate.

::这是准确的。

Review
::回顾

Solve the following equations. Write your answer in the form: $\begin{aligned} Variable = \underline{} . \end{aligned}$
::解决以下方程式。以窗体写您的回答: 变量 = _ 。

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

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

$\begin{aligned} \frac{x}{9} = 9 \end{aligned}$
::9x9=9

$\begin{aligned} \frac{x}{5} = 3 \end{aligned}$
::5 x 5 = 3

$\begin{aligned} \frac{20}{y} = 4 \end{aligned}$
::20y=4

$\begin{aligned} \frac{x}{2} = 18 \end{aligned}$
::x 2 = 18

$\begin{aligned} \frac{x}{3} = 2 \end{aligned}$
::x 3 = 2

$\begin{aligned} \frac{x}{12} = 9 \end{aligned}$
::12x12=9

$\begin{aligned} \frac{x}{14} = 3 \end{aligned}$
::14 x 14 = 3

$\begin{aligned} \frac{x}{11} = 9 \end{aligned}$
::11x11=9

$\begin{aligned} \frac{x}{13} = 4 \end{aligned}$
::13x13=4

$\begin{aligned} \frac{12}{x} = 3 \end{aligned}$
::12 x = 3

$\begin{aligned} \frac{44}{x} = 4 \end{aligned}$
::44x=4

$\begin{aligned} \frac{32}{x} = 8 \end{aligned}$
::32x=8

$\begin{aligned} \frac{90}{x} = 30 \end{aligned}$
::90x=30

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

单变量除法方程

Solving Single-Variable Division Equations ::解决单一可变分区

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 Division Equations
::解决单一可变分区

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