Section: 分数方程中加法的逆性质 | 3.9分数方程中加法的逆性质

Section outline

Cody is making a pie for his math class for pi day. His mom gives him a big bowl of blueberries and a package of blueberries to get started. Cody pours the blueberries from the package into a measuring cup and sees that it is $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ cups of blueberries. He adds these blueberries to the big bowl. He is about to start adding the rest of the ingredients when he realizes he should know how many cups of blueberries he has! He measures all the blueberries in the bowl and finds that he has $\begin{aligned} 6 \frac{1}{4} \end{aligned}$ cups of blueberries. How can Cody figure out how many blueberries were in the bowl to start with?
::Cody把蓝莓倒进一个量度杯中, 并看到它有112杯蓝莓。他把这些蓝莓加到一个大的碗里。当他意识到自己应该知道有多少杯蓝莓的时候, 他就要开始添加其余的成分了! 他测量了碗里所有的蓝莓, 发现他有64杯蓝莓。科迪怎么才能找出碗里有多少蓝莓要开始呢?

In this concept, you will learn how to apply the inverse property of addition to solve fraction equations.
::在此概念中,您将学会如何应用附加的反属性来解析分数方程式。

Applying the Inverse Property of Addition in Fraction Equations
::应用 " 分数等量中增加的反向属性 "

Recall that the inverse property of addition states that the sum of any number and its opposite is zero. In symbols, it says that for any number $\begin{aligned} a \end{aligned}$ :
::回顾增加的反面属性指出,任何数字之和及其对数之和为零。在符号中,它说,对于任何数字a:

$\begin{aligned} a + (- a) = 0 \end{aligned}$

You can use the inverse property of addition to help you to solve equations. Remember that when solving an equation you want to isolate the variable which means you want to get the variable by itself on one side of the equation. Sometimes, when you add or subtract the same amount from both sides of the equation you will be able to isolate the variable with the help of the inverse property of addition.
::您可以使用附加的反向属性来帮助您解析方程式。记住, 在解析方程式时, 您想要分离变量, 这意味着您想要在方程式的一边自己获取变量。有时, 当您从方程式的两边增减相同数量时, 您可以在反向附加属性的帮助下, 将变量分离出来。

Here is an example.
::举一个例子。

Solve the following equation for $\begin{aligned} x \end{aligned}$ .
::解决 x 的以下方程式。

$\begin{aligned} x + 1 \frac{1}{2} = 3 \frac{3}{4} \end{aligned}$

First, notice that $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ is added to $\begin{aligned} x \end{aligned}$ on one side of the equation. You want to isolate the $\begin{aligned} x \end{aligned}$ . Subtract $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ from both sides of the equation.
::首先,请注意方程的一面 x 中添加了 11 2。您想要从方程的两侧分离 x 。减号 1 12 2 。

$\begin{aligned} x + 1 \frac{1}{2} - 1 \frac{1}{2} = 3 \frac{3}{4} - 1 \frac{1}{2} \end{aligned}$

Next, simplify the left side of the equation by combining like terms . $\begin{aligned} 1 \frac{1}{2} - 1 \frac{1}{2} \end{aligned}$ makes 0 which leaves the $\begin{aligned} x \end{aligned}$ by itself.
::接下来,简化方程的左侧,将类似术语合并。 1,1、2 - 1,1,2 使 X 本身为0 。

$\begin{aligned} x = 3 \frac{3}{4} - 1 \frac{1}{2} \end{aligned}$

Now, simplify the right side of the equation by combining like terms. Use what you have learned about mixed number subtraction . You will need to find a common denominator and then convert to improper fractions first.
::现在, 简化方程的右侧, 将类似条件合并。使用您学到的混合数字减法。您需要先找到一个共同的分母, 然后转换成不适当的分数。

$\begin{aligned} \begin{array}{rcl} x & = & 3 \frac{3}{4} - 1 \frac{2}{4} \\ x & = & \frac{15}{4} - \frac{6}{4} \\ x & = & \frac{9}{4} = 2 \frac{1}{4} \end{array} \end{aligned}$

The answer is $\begin{aligned} x = 2 \frac{1}{4} \end{aligned}$ .
::答案是x=2 1 4。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Cody and his blueberry pie.
::之前,你得到一个问题关于科迪和他的蓝莓派。

He has a big bowl filled with blueberries. He's not sure how many cups of blueberries were in the bowl to start with, but after adding $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ cups of blueberries the bowl has $\begin{aligned} 6 \frac{1}{4} \end{aligned}$ cups of blueberries in it. Cody wants to know how many cups of blueberries were in the bowl originally.
::他有一个大碗里装满了蓝莓。他不知道碗里装了多少杯蓝莓, 但是在加了12杯蓝莓之后,碗里装了614杯蓝莓。Cody想知道碗里原来装了多少杯蓝莓。

Cody could set up an equation to represent this situation. He doesn't know how many cups of blueberries were in the bowl originally, so that can be his variable. Let $\begin{aligned} x \end{aligned}$ be the original number of cups of blueberries in the bowl.
::科迪可以设置一个方程式来代表这种情况。他不知道最初碗里有多少杯蓝莓, 所以这可能是他的变量。让 x 成为碗里最初的蓝莓杯数。

Cody knows that the original number of cups of blueberries plus the $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ cups of blueberries he added will give him the $\begin{aligned} 6 \frac{1}{4} \end{aligned}$ cups of blueberries in the bowl now. He can turn this into an equation.
::科迪知道最初的蓝莓杯数加上他加起来的12杯蓝莓将给他碗里的614杯蓝莓

$\begin{aligned} x + 1 \frac{1}{2} = 6 \frac{1}{4} \end{aligned}$

Now, he can solve this equation by isolating the $\begin{aligned} x \end{aligned}$ . He can subtract $\begin{aligned} 1 \frac{1}{2} \end{aligned}$ from both sides of the equation.
::现在,他可以通过分离 x 来解答这个方程式。他可以从方程式的两侧减去112。

$\begin{aligned} x + 1 \frac{1}{2} - 1 \frac{1}{2} = 6 \frac{1}{4} - 1 \frac{1}{2} \end{aligned}$

Next, he can simplify the left side of the equation by combining like terms. $\begin{aligned} 1 \frac{1}{2} - 1 \frac{1}{2} \end{aligned}$ makes 0 which leaves the $\begin{aligned} x \end{aligned}$ by itself.
::其次,他可以将类似术语合并,简化方程的左侧。 1,12-1,12使0使x本身留下。

$\begin{aligned} x = 6 \frac{1}{4} - 1 \frac{1}{2} \end{aligned}$

Now, he can simplify the right side of the equation by combining like terms. He will need to find a common denominator and then convert to improper fractions first.
::现在,他可以通过将类似条件合并来简化方程式的右侧。他需要找到一个共同的分母,然后首先转换成不适当的分数。

$\begin{aligned} \begin{array}{rcl} x & = & 6 \frac{1}{4} - 1 \frac{2}{4} \\ x & = & \frac{25}{4} - \frac{6}{4} \\ x & = & \frac{19}{4} = 4 \frac{3}{4} \end{array} \end{aligned}$

The answer is the bowl had $\begin{aligned} 4 \frac{3}{4} \end{aligned}$ cups of blueberries in it originally.
::答案是碗里有4 3 4杯蓝莓

Example 2
::例2

Solve the following equation for $\begin{aligned} x \end{aligned}$ .
::解决 x 的以下方程式。

$\begin{aligned} x + \frac{2}{3} = \frac{5}{6} \end{aligned}$

First, notice that $\begin{aligned} \frac{2}{3} \end{aligned}$ is added to $\begin{aligned} x \end{aligned}$ on one side of the equation. To isolate the $\begin{aligned} x \end{aligned}$ , subtract $\begin{aligned} \frac{2}{3} \end{aligned}$ from both sides of the equation.
::首先, 注意方程的一面将 2 3 添加到 x 上。要孤立 x, 请从方程的两侧减去 2 3 。

$\begin{aligned} x + \frac{2}{3} - \frac{2}{3} = \frac{5}{6} - \frac{2}{3} \end{aligned}$

Next, simplify the left side of the equation by combining like terms. $\begin{aligned} \frac{2}{3} - \frac{2}{3} \end{aligned}$ makes 0 which leaves the $\begin{aligned} x \end{aligned}$ by itself.
::接下来,简化方程的左侧,将类似术语合并。 2 3 - 2 3 使 0 本身留下 x 。

$\begin{aligned} x = \frac{5}{6} - \frac{2}{3} \end{aligned}$

Now, simplify the right side of the equation by combining like terms. Use what you have learned about fraction subtraction. You will need to find a common denominator first.
::现在, 简化方程的右侧, 将类似条件合并。使用您学到的关于分数减法的知识。您需要先找到一个共同的分母。

$\begin{aligned} \begin{array}{rcl} x & = & \frac{5}{6} - \frac{4}{6} \\ x & = & \frac{1}{6} \end{array} \end{aligned}$

The answer is $\begin{aligned} x = \frac{1}{6} \end{aligned}$ .
::答案是 x = 1 6 。

Example 3
::例3

Solve the following equation for $\begin{aligned} x \end{aligned}$ .
::解决 x 的以下方程式。

$\begin{aligned} \frac{1}{7} + x = \frac{5}{7} \end{aligned}$

First, isolate the $\begin{aligned} x \end{aligned}$ by subtracting $\begin{aligned} \frac{1}{7} \end{aligned}$ from both sides of the equation.
::首先,从方程两侧减去17,将x分离出来。

$\begin{aligned} \frac{1}{7} + x - \frac{1}{7} = \frac{5}{7} - \frac{1}{7} \end{aligned}$

Next, simplify the left side of the equation by combining like terms. $\begin{aligned} \frac{1}{7} - \frac{1}{7} \end{aligned}$ makes 0 which leaves the $\begin{aligned} x \end{aligned}$ by itself.
::接下来,简化方程的左侧,将类似术语合并。 1 7 - 1 7 使 0 使x 本身留下。

$\begin{aligned} x = \frac{5}{7} - \frac{1}{7} \end{aligned}$

Now, simplify the right side of the equation by combining like terms. Use what you have learned about fraction subtraction.
::现在, 简化方程的右侧, 将类似条件合并。使用您学到的关于分数减法的知识。

$\begin{aligned} x = \frac{4}{7} \end{aligned}$

The answer is $\begin{aligned} x = \frac{4}{7} \end{aligned}$ .
::答案是x=4 7。

Example 4
::例4

Solve the following equation for $\begin{aligned} y \end{aligned}$ .
::解决y的以下方程式。

$\begin{aligned} \frac{1}{3} + y = \frac{5}{6} \end{aligned}$

First, isolate the $\begin{aligned} y \end{aligned}$ by subtracting $\begin{aligned} \frac{1}{3} \end{aligned}$ from both sides of the equation.
::首先,从方程的两侧减去13,将y分离出来。

$\begin{aligned} \frac{1}{3} + y - \frac{1}{3} = \frac{5}{6} - \frac{1}{3} \end{aligned}$

Next, simplify the left side of the equation by combining like terms. $\begin{aligned} \frac{1}{3} - \frac{1}{3} \end{aligned}$ makes 0 which leaves the $\begin{aligned} y \end{aligned}$ by itself.
::接下来,简化方程的左侧,将类似术语合并。 1 3 - 1 3 使 0 自动离开 y 。

$\begin{aligned} y = \frac{5}{6} - \frac{1}{3} \end{aligned}$

Now, simplify the right side of the equation by combining like terms. Use what you have learned about fraction subtraction. You will need to find a common denominator first. Don't forget to simplify your answer.
::现在, 简化方程的右侧, 将类似条件合并。使用您学到的关于分数减法的知识。您需要先找到一个共同的分母。不要忘了简化您的答案。

$\begin{aligned} \begin{array}{rcl} y & = & \frac{5}{6} - \frac{2}{6} \\ x & = & \frac{3}{6} = \frac{1}{2} \end{array} \end{aligned}$

The answer is $\begin{aligned} x = \frac{1}{2} \end{aligned}$ .
::答案是 x = 1 2 。

Example 5
::例5

Solve the following equation for $\begin{aligned} y \end{aligned}$ .
::解决y的以下方程式。

$\begin{aligned} y + \frac{1}{2} = \frac{5}{6} \end{aligned}$

First, isolate the $\begin{aligned} y \end{aligned}$ by subtracting $\begin{aligned} \frac{1}{2} \end{aligned}$ from both sides of the equation.
::首先,从方程的两侧减去12,将y分离出来。

$\begin{aligned} y + \frac{1}{2} - \frac{1}{2} = \frac{5}{6} - \frac{1}{2} \end{aligned}$

Next, simplify the left side of the equation by combining like terms. $\begin{aligned} \frac{1}{2} - \frac{1}{2} \end{aligned}$ makes 0 which leaves the $\begin{aligned} y \end{aligned}$ by itself.
::接下来,简化方程的左侧, 将类似条件合并。 1 2 - 1 2 使 0 本身离开 y 。

$\begin{aligned} y = \frac{5}{6} - \frac{1}{2} \end{aligned}$

Now, simplify the right side of the equation by combining like terms. Use what you have learned about fraction subtraction. You will need to find a common denominator first. Don't forget to simplify your answer.
::现在, 简化方程的右侧, 将类似条件合并。使用您学到的关于分数减法的知识。您需要先找到一个共同的分母。不要忘了简化您的答案。

$\begin{aligned} \begin{array}{rcl} y & = & \frac{5}{6} - \frac{3}{6} \\ y & = & \frac{2}{6} = \frac{1}{3} \end{array} \end{aligned}$

The answer is $\begin{aligned} x = \frac{1}{3} \end{aligned}$ .
::答案是 x = 1 3 。

Review
::回顾

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

$\begin{aligned} x + \frac{2}{5} = \frac{5}{5} \end{aligned}$
::x + 2 + 2 5 = 5 5 5

$\begin{aligned} x + \frac{2}{8} = \frac{6}{8} \end{aligned}$
::x + 2 8 = 6 8

$\begin{aligned} x + \frac{3}{9} = \frac{4}{9} \end{aligned}$
::x + 3 9 = 4 9

$\begin{aligned} x + \frac{5}{7} = \frac{6}{7} \end{aligned}$
::x + 5 7 = 6 7

$\begin{aligned} x + \frac{10}{12} = \frac{11}{12} \end{aligned}$
::x + 10 12 = 11 12x + 10 12 = 11 12

$\begin{aligned} x + \frac{3}{15} = \frac{10}{15} \end{aligned}$
::x + 3 15 = 10 15

$\begin{aligned} x + \frac{9}{13} = \frac{12}{13} \end{aligned}$
::x + 9 9 13 = 12 13

$\begin{aligned} x + \frac{6}{14} = \frac{12}{14} \end{aligned}$
::x + 6 6 14 = 12 14

$\begin{aligned} x + \frac{1}{5} = \frac{6}{10} \end{aligned}$
::x + 1 5 = 6 10x + 1 5 = 6 10

$\begin{aligned} x + \frac{1}{2} = \frac{11}{12} \end{aligned}$
::x + 1 2 = 11 12

$\begin{aligned} x - 1 \frac{1}{2} = 4 \end{aligned}$
::x - 1 1 1 2 = 4

$\begin{aligned} 2 \frac{1}{4} + x = 3 \frac{3}{4} \end{aligned}$
::2 1 4 + x = 3 3 4

$\begin{aligned} x - \frac{7}{8} = 2 \frac{3}{4} \end{aligned}$
::x - 7 8 = 2 3 4

Ludmilla, Brent, and Rudy have $\begin{aligned} 8 \frac{5}{6} \end{aligned}$ feet of taffy that they have to sell to raise money for the school drama club. Brent has already sold $\begin{aligned} 3 \frac{2}{3} \end{aligned}$ feet of taffy and Rudy plans to sell exactly $\begin{aligned} 2 \frac{3}{4} \end{aligned}$ feet. How much taffy does Ludmilla have to sell, if they sell all of the taffy?
::布伦特已经卖掉了323英尺的塔菲,鲁迪打算卖掉234英尺。如果卢德米拉卖掉所有的塔菲,他们要卖多少塔菲?

Ron, Jung-Ho, and Sarah have a lawn mowing business. Today they are cutting an enormous lawn. Sarah agrees to start and will mow $\begin{aligned} \frac{3}{8} \end{aligned}$ of the lawn. Ron will only mow $\begin{aligned} \frac{1}{7} \end{aligned}$ of the lawn, but he's willing to work during the hottest time of the day. How much of the lawn is Jung-Ho responsible for completing?
::Ron, Jung-Ho和Sarah有割草场的生意。今天,他们正在砍伐一个巨大的草坪。Sarah同意开始并修剪3-8片草坪。Ron只修剪1-7片草坪,但他愿意在一天最热的时候工作。有多少草坪由Jung-Ho负责完成?

Review (Answers)
::回顾(答复)

Section outline

Applying the Inverse Property of Addition in Fraction Equations ::应用 " 分数等量中增加的反向属性 "

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)

Applying the Inverse Property of Addition in Fraction Equations
::应用 " 分数等量中增加的反向属性 "

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)