Section: 单变量乘法方程 | 12.8单变量乘法方程

Section outline

Mrs. Jackson’s class has done well during the first half of the school’s field day events. One of her students won the 100-meter dash and another placed second in the high jump. Now, the students are hot and thirsty. Twelve of the students give Mrs. Jackson money to buy them a drink from the concession stand. The total for the 12 drinks is $21. When she returns, she passes out the drinks and, because she teaches math, asks the students to use a single variable multiplication equation to figure out how much each drink cost. That way, she will know how much change to give each of them. Can you figure out how much each cost?
::Jackson夫人的班级在学校的田间活动前半段期间表现良好,她的学生中有一个赢得了100米的破折线,另一个在跳跃中排第二位。现在,学生们很热,很渴。12名学生给Jackson夫人钱,从特许摊位请他们喝杯酒。12杯酒的总数是21美元。当她回来时,她喝完酒,并且因为她教数学,她要求学生们用一个可变乘数方程式来计算每个饮料的费用。这样,她就会知道给每个人多少变化。你能弄清楚每杯多少钱吗?

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

Solving Single-Variable Multiplication Equations
::解决单变数乘数等数

Here is a multiplication equation with a single variable $\begin{aligned} x \end{aligned}$ .
::这里是一个乘法方程式, 只有一个变量 x 。

$\begin{aligned} 5 x = 30 \end{aligned}$

You can figure out the value of $\begin{aligned} x \end{aligned}$ in two ways.
::您可以以两种方式找出 x 的值。

Use mental math
::使用心理数学

Use the inverse operation
::使用反向操作

To use mental math, ask yourself, “What times five is equal to thirty?” Then, using the times table, you can figure out that 5 times 6 is equal to thirty. So, the value of $\begin{aligned} x \end{aligned}$ is 6.
::使用心理数学,请问自己,“什么乘以5等于30?” 然后,用时间表,你可以发现5乘以6等于30。那么,x值是6。

To use the inverse operation, use the opposite operation of multiplication, which is division .
::使用反向操作,使用反向的乘法操作,即除法。

First, get the variable alone on one side of the equation by dividing both sides by the number next to the variable. In this example, divide both sides by 5.
::首先, 通过将两边除以变量旁边的数, 将变量单方方程式的一方获得变量。在此示例中, 将两边除以 5 。

$\begin{aligned} \frac{5 x}{5} = \frac{30}{5} \end{aligned}$

The fives cancel each other out because five divided by five is one, and “ $\begin{aligned} x \end{aligned}$ ” times 1 is “ $\begin{aligned} x \end{aligned}$ ”. On the right side, thirty divided by five is six.
::五除五除五等于一,“x”乘1等于“x”。在右侧,三十除五等于六。

$\begin{aligned} \begin{array}{rcl} \frac{5 x}{5} & = & \frac{30}{5} \\ x & = & 6 \end{array} \end{aligned}$

You can check your work by substituting the value of $\begin{aligned} x \end{aligned}$ back into the original equation. If both sides are equal, then your work is accurate and correct.
::您可以将 x 的值重置到原始方程中, 查看您的工作。如果两边相等, 那么您的工作是准确和正确的。

$\begin{aligned} \begin{array}{rcl} 5 x & = & 30 \\ 5 (6) & = & 30 \\ 30 & = & 30 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Here is another example.
::下面是另一个例子。

$\begin{aligned} 7 y = 49 \end{aligned}$

First, use the inverse operation of multiplication and divide both sides by 7 to get the variable alone.
::首先,利用乘法的反向操作,将两边除以7, 以单独获得变量。

$\begin{aligned} \frac{7 y}{7} = \frac{49}{7} \end{aligned}$

Next, the 7’s cancel each other out, leaving $\begin{aligned} y \end{aligned}$ on one side of the equation. On the other side, divide 49 by 7. You are left with $\begin{aligned} y = 7 \end{aligned}$ .
::接下来,7 的相互取消,将你们留在方程的一边。在另一边,49除以7,剩下的是y=7。

$\begin{aligned} \begin{array}{rcl} \frac{7 y}{7} & = & \frac{49}{7} \\ y & = & 7 \end{array} \end{aligned}$

Then, check your work by substituting 7 back into the original problem for $\begin{aligned} y \end{aligned}$ .
::然后,检查你的工作通过替换7 回到原来的问题y。

$\begin{aligned} \begin{array}{rcl} 7 y & = & 49 \\ 7 (7) & = & 49 \\ 49 & = & 49 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Mrs. Jackson and her students looking for relief from the heat.
::早些时候,你遇到了一个问题杰克逊夫人和她的学生想要从炎热中解脱出来

Mrs. Jackson bought 12 drinks priced the same and the total cost was $21. How can she use a single variable multiplication equation to figure out how much each drink cost?
::Jackson夫人买了12杯饮料,价格相同,总费用为21美元。她如何使用一个可变乘法方程式来计算每杯饮料的费用?

First, write an equation to represent the situation.
::首先,写一个方程式来代表局势。

12 times a number (price of drink) is 21.
::12乘以一个数字(饮料价格)为21。

$\begin{aligned} 12 y = 21 \end{aligned}$

Next, use the inverse operation of multiplication and divide both sides by 12 to get the variable alone.
::其次,使用乘法的反向操作,将两边除以12,以单独获得变量。

$\begin{aligned} \frac{12 y}{12} = \frac{21}{12} \end{aligned}$

The 12’s cancel each other out, leaving $\begin{aligned} y \end{aligned}$ alone on one side of the equal sign. On the other side, divide 21 by 12 to get $\begin{aligned} y = 1.75 \end{aligned}$ .
::12人相互勾销,任由你独自站在平等标志的一边。在另一边,21除以12以获得y = 1.75。

$\begin{aligned} \begin{array}{rcl} \frac{12 y}{12} & = & \frac{21}{12} \\ y & = & 1.75 \end{array} \end{aligned}$

Then, check your work by substituting 1.75 back into the original problem for $\begin{aligned} y \end{aligned}$ .
::然后,检查你的工作通过替换 1.75 回到原来的问题y。

$\begin{aligned} \begin{array}{rcl} 12 y & = & 21 \\ 12 (1.75) & = & 21 \\ 21 & = & 21 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

The price of each drink is $1.75.
::每杯酒的价格为1.75美元。

Solve the following equations and write your answer in the form $\begin{aligned} Variable = \underline{} \end{aligned}$ .
::解决以下方程式,并以变量 = _ 的形式写入您的回答。

Example 2
::例2

$\begin{aligned} 12 y = 84 \end{aligned}$

First, use the inverse operation of multiplication and divide both sides by 12 to get the variable alone.
::首先,利用乘法的反向操作,将两边除以12,以单独获得变量。

$\begin{aligned} \frac{12 y}{12} = \frac{84}{12} \end{aligned}$

Next, the 12’s cancel each other out, leaving $\begin{aligned} y \end{aligned}$ alone. Dividing 84 by 12 leaves $\begin{aligned} y = 7 \end{aligned}$ .
::其次,12人相互取消,留下你们一人。84人分离开12个假y=7。

$\begin{aligned} \begin{array}{rcl} \frac{12 y}{12} & = & \frac{84}{12} \\ y & = & 7 \end{array} \end{aligned}$

Then, check your work by substituting 7 back into the original problem for $\begin{aligned} y \end{aligned}$ .
::然后,检查你的工作通过替换7 回到原来的问题y。

$\begin{aligned} \begin{array}{rcl} 12 y & = & 84 \\ 12 (7) & = & 84 \\ 84 & = & 84 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Example 3
::例3

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

First, use the inverse operation of multiplication and divide both sides by 8 to get the variable alone.
::首先,利用乘法的反向操作,将两边除以8,以单独获得变量。

$\begin{aligned} \frac{8 x}{8} = \frac{64}{8} \end{aligned}$

Next, the 8’s cancel each other out, leaving $\begin{aligned} x \end{aligned}$ alone. Dividing 64 by 8 leaves $\begin{aligned} x = 8 \end{aligned}$ .
::接下去,8人相互勾销,留下x一人。64人乘8个假x=8。

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

Then, check your work by substituting 8 back into the original problem for $\begin{aligned} x \end{aligned}$ .
::然后,检查你的工作,替换 8回到原问题xx。

$\begin{aligned} \begin{array}{rcl} 8 x & = & 64 \\ 8 (8) & = & 64 \\ 64 & = & 64 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Example 4
::例4

$\begin{aligned} 2 a = 26 \end{aligned}$

First, use the inverse operation of multiplication and divide both sides by 2 to get the variable alone.
::首先,利用乘法的反向操作,将两边除以2,以单独获得变量。

$\begin{aligned} \frac{2 a}{2} = \frac{26}{2} \end{aligned}$

Next, the 2’s cancel each other out, leaving $\begin{aligned} a \end{aligned}$ alone. So, divide 26 by 2 to get $\begin{aligned} a = 13 \end{aligned}$ .
::其次,二人相互勾销,留下一个人。因此,将26除以2分成13。

$\begin{aligned} \begin{array}{rcl} \frac{2 a}{2} & = & \frac{26}{2} \\ a & = & 13 \end{array} \end{aligned}$

Then, check your work by substituting 13 back into the original problem for $\begin{aligned} a \end{aligned}$ .
::然后,检查你的工作通过替换13 回到原来的问题

$\begin{aligned} \begin{array}{rcl} 2 a & = & 26 \\ 2 (13) & = & 26 \\ 26 & = & 26 \end{array} \end{aligned}$

The answer is correct.
::答案是正确的。

Example 5
::例5

$\begin{aligned} 6 y = 42 \end{aligned}$

First, use the inverse operation of multiplication and divide both sides by 6 to get the variable alone.
::首先,利用乘法的反向操作,将两边除以6,以单独获得变量。

$\begin{aligned} \frac{6 y}{6} = \frac{42}{6} \end{aligned}$

Next, the 6’s cancel each other out, leaving $\begin{aligned} y \end{aligned}$ alone. Dividing 42 by 6 leaves $\begin{aligned} y = 7 \end{aligned}$ .
::之后,6人互相取消,离开你一人。 42人乘6人乘Y=7。

$\begin{aligned} \begin{array}{rcl} \frac{6 y}{6} & = & \frac{42}{6} \\ y & = & 7 \end{array} \end{aligned}$

Then, check your work by substituting 7 back into the original problem for $\begin{aligned} y \end{aligned}$ .
::然后,检查你的工作通过替换7 回到原来的问题y。

$\begin{aligned} \begin{array}{rcl} 6 y & = & 42 \\ 6 (7) & = & 42 \\ 42 & = & 42 \end{array} \end{aligned}$

The answer is correct.

::答案是正确的。

Review
::回顾

Solve each single-variable multiplication equation. Write your answer in the form $\begin{aligned} Variable = \underline{} \end{aligned}$ .
::解析每个单变量乘法方程式。将答案写成“变量” = _。

$\begin{aligned} 7 y = 14 \end{aligned}$
::7y = 14

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

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

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

$\begin{aligned} 3 y = 12 \end{aligned}$
::3y = 12

$\begin{aligned} 8 a = 72 \end{aligned}$
::8 a = 72

$\begin{aligned} 12 v = 36 \end{aligned}$
::12 v = 36

$\begin{aligned} 9 x = 45 \end{aligned}$
::9x=45

$\begin{aligned} 10 y = 100 \end{aligned}$
::10y = 100

$\begin{aligned} 7 x = 21 \end{aligned}$
::7x=21

$\begin{aligned} 9 a = 99 \end{aligned}$
::9 a = 99

$\begin{aligned} 16 x = 32 \end{aligned}$
::16 x = 32

$\begin{aligned} 14 y = 28 \end{aligned}$
::14y = 28

$\begin{aligned} 13 y = 39 \end{aligned}$
::13y = 39

$\begin{aligned} 7 y = 140 \end{aligned}$
::7y = 140

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

Solving Single-Variable Multiplication Equations ::解决单变数乘数等数

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

Solving Single-Variable Multiplication Equations
::解决单变数乘数等数

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