节: 使用交叉乘法解决合理等式 | 9.7 利用交叉乘法解决合理等式

章节大纲

A doctor wants to prescribe the antibiotic amoxicillin to a 2-year-old child for an infection. If the child weighs 40 pounds, and the dose that is given twice a day is 20 mg of amoxicillin per kilogram of body weight, how much amoxicillin should the child receive with each dose?
::医生想将抗生素氨基西林开给一个2岁儿童进行感染。如果孩子体重为40磅,而且每天两次的剂量为每公斤体重20毫克甲基西林,那么孩子每服一剂就能得到多少氨基西林?

We can set up a proportion , a type of rational equation , to solve this. We discuss the approach in this section.
::我们可以建立一个比例,一种理性的等式,来解决这个问题。我们在本节中讨论这个方法。

Solving Rational Equations by Cross-Multiplication
::通过交叉重复解决理性等号

In this section, we will use a technique called cross- multiplication , which says we can multiply the numerator of one fraction with denominator of the other fraction on the opposite side of the equal sign.
::在本节中,我们将使用一种叫作交叉乘法的技术,它表示我们可以将一个分数的分子乘以相同符号对面另一个分母的分母。

Cross-Multiplication
::交叉重复

For any expressions A , B , C , and D : $\begin{align*}\frac{A}{B}=\frac{C}{D}\end{align*}$ , then $\begin{align*}AD=BC\end{align*}$ .

Why does this work? Let's multiply by the LCD as we did in the previous section.
::为什么这样工作?让我们像上一节那样,用液晶体成倍地乘以液晶。

$\begin{aligned} \frac{A}{B} & = \frac{C}{D} \\ B D \cdot \frac{A}{B} & = \frac{C}{D} \cdot B D \\ A D & = B C \end{aligned}$
$\begin{align*}\frac{A}{B}&=\frac{C}{D} \\ \\ \cancel{B}D \cdot \frac{A}{\cancel{B}}&= \frac{C}{\cancel{D}} \cdot B\cancel{D} \\ \\ AD&=BC\end{align*}$
::AB=CDBDAB=CD&BDAD=BC =CDB=CDBDAB=CD&BDAD=BC =CDBB=CDDD=CD&BDDAD=BC =BC=CDDD=BCDD=BBD=BBDAD=BCD=BBD=BBD=BBD=BBD=BD=BBD=BBD=BBD=BBD=BBD=BBB=BBB=BBBB=BBB=BBBB=BB=BD=BB=BBB=BBBB=BB=BD=BBD=B=BBDAB=B=BDAB=BBD=BBD=BD=BBD=BDBBD=BD=BD=BD=BD=BD=BD=BD=BD=B=BD=BD=BD=BD=BDBD=BD=B=B=BD=BD=B=B=B=B=B=BD=BDDD=BD=BD=BD=BDD=BD=BD=BD=BD=B=B=BD=BD=BD=B=B=BD=B=BD=BD=BD=BD=BD=B=BD=BD=B=B=B=BD=BD=B=BD=BD=BD=BD=BD=BD=B=BBBD=BD=BD=B=BBBBBBD=BD=BBBBBBBBBBD=BD=BD=BD=B=B=B=B=B=B=B=B=B=BBBBBBBBD=B=BD=B=B=BBBB=B=B=B=B=B=B=B=B=B=B=BBBBBBBBBB

Here is an example of a proportion , one rational expression set equal to another rational expression, that we can solve using cross-multiplication.
::这里有一个比例的例子,一个合理表达方式与另一个合理表达方式相当,我们可以使用交叉乘法加以解决。

Example 1
::例1

Solve $\begin{align*}\frac{x}{2x-3}=\frac{3x}{x+11}\end{align*}$ .
::溶解 x2x-3=3xx+11 。

Solution: Use cross-multiplication to solve the problem. You can use the example above as a guideline.
::解决方案: 使用交叉倍数来解决问题。您可以使用上面的示例作为指南。

Check your answers. It is possible to get extraneous solutions with rational equations.
::检查您的答案。可以用理性方程式获得不相干的解决办法。

$\begin{aligned} \frac{0}{2 \cdot 0 - 3} & = \frac{3 \cdot 0}{0 + 11} & \frac{4}{2 \cdot 4 - 3} = \frac{3 \cdot 4}{4 + 11} \\ \frac{0}{- 3} & = \frac{0}{11} & \frac{4}{5} = \frac{12}{15} \\ 0 & = 0 & \frac{4}{5} = \frac{4}{5} \end{aligned}$
$\begin{align*}\frac{0}{2 \cdot 0-3}&=\frac{3 \cdot 0}{0+11} && \frac{4}{2 \cdot 4-3}=\frac{3 \cdot 4}{4+11} \\ \frac{0}{\text{-}3}&=\frac{0}{11} \ && \qquad \quad \frac{4}{5}=\frac{12}{15} \\ 0&=0 && \qquad \quad \frac{4}{5}=\frac{4}{5}\end{align*}$

Example 2
::例2

Solve $\begin{align*}\frac{x^2}{2x-5}=\frac{x+8}{2}\end{align*}$ .
::解决 x22x- 5=x+82。

Solution: Cross-multiply.
::解决方案:交叉倍增。

$\begin{aligned} \frac{x^{2}}{2 x - 5} & = \frac{x + 8}{2} \\ 2 x^{2} + 11 x - 40 & = 2 x^{2} \\ 11 x - 40 & = 0 \\ 11 x & = 40 \\ x & = \frac{40}{11} \end{aligned}$
$\begin{align*}\frac{x^2}{2x-5}&=\frac{x+8}{2} \\ 2x^2+11x-40&=2x^2 \\ 11x-40&=0 \\ 11x&=40 \\ x&=\frac{40}{11}\end{align*}$
::x22x- 5=x+822x2+11x- 40=2x221x- 40=011x=011x=40x=4011

Check the answer: $\begin{align*}\frac{\left(\frac{40}{11}\right)^2}{\frac{80}{11}-5}=\frac{\frac{40}{11}+8}{2} \rightarrow \frac{1,\!600}{121} \div \frac{25}{11}=\frac{128}{11} \div 2 \rightarrow \frac{64}{11}=\frac{128}{22}\end{align*}$ .
::答: (4011) 28011- 5= 4011+821 6001212511=1281126411=12822。

by Mathispower4u demonstrates how to solve rational equations.
::Mathispower4u 演示如何解析理性方程式。



Example 3
::例3

Solve $\begin{align*}\frac{9-x}{x^2}=\frac{4}{3x}\end{align*}$ .
::解决 9 - 2x2=43x。

Solution:

$\begin{aligned} \frac{9 - x}{x^{2}} & = \frac{4}{- 3 x} \\ 4 x^{2} & = - 27 x + 3 x^{2} \\ x^{2} + 27 x & = 0 \\ x (x + 27) & = 0 \\ x & = 0 and - 27 \end{aligned}$
$\begin{align*}\frac{9-x}{x^2}&=\frac{4}{\text{-}3x} \\ 4x^2&=\text{-}27x+3x^2 \\ x^2+27x&=0 \\ x(x+27)&=0 \\ x&=0 \ \text{and} \ \text{-}27 \end{align*}$
::溶解度: 9-xx2=4-3x4x2=27x3x2+27x0x(x+27)=0x=0和-27

$\begin{aligned} Check : x = 0 \to \frac{9 - 0}{0^{2}} & = \frac{4}{- 3 (0)} & x = - 27 \to \frac{9 + 27}{{(- 27)}^{2}} = \frac{4}{- 3 (- 27)} \\ u n d & = u n d & \frac{36}{729} = \frac{4}{81} \\ \frac{4}{81} = \frac{4}{81} \end{aligned}$
$\begin{align*}\text{Check}: x=0 \rightarrow \frac{9-0}{0^2}&=\frac{4}{\text{-}3 \left(0\right)} && x=-27 \rightarrow \frac{9+27}{\left(\text{-}27\right)^2}=\frac{4}{\text{-}3 \left(\text{-}27\right)} \\ und&=und && \qquad \qquad \qquad \quad \frac{36}{729}=\frac{4}{81} \\ & && \qquad \qquad \qquad \quad \ \frac{4}{81}=\frac{4}{81}\end{align*}$
::检查:x=09=002-002=4-3(0)x_27_9+27(-272)=4-3(-27und=und36729=481481=481=481

$\begin{align*}x = 0\end{align*}$ is an extraneous solution .
::x=0 是一个不相干溶液。

Example 4
::例4

A doctor wants to prescribe the antibiotic amoxicillin to a 2-year-old child for an infection. If the child weighs 40 pounds, and the dose that is given twice a day is 20 mg of amoxicillin per kilogram of body weight, how much amoxicillin should the child receive with each dose?
::医生想将抗生素氨基西林开给一个2岁儿童进行感染。如果孩子体重为40磅,而且每天两次的剂量为每公斤体重20毫克甲基西林,那么孩子每服一剂就能得到多少氨基西林?

Solution: First, we need to convert the child's body weight to kilograms, so that we use the same units. One pound is equal to 0.454 kilograms. A 40-pound child is then
::解决办法:首先,我们需要将儿童的身体重量转换成公斤,以便使用相同的单位。一磅等于0.454公斤。然后,一个40磅的婴儿

$\begin{aligned} 40 lb \cdot \frac{0.454 kg}{1 lb} = 18.144 kg . \end{aligned}$
$\begin{align*}40 \ \cancel{\text{lb}} \cdot \frac{0.454 \ \text{kg}}{1 \ \cancel{\text{lb}}}=18.144 \ \text{kg}.\end{align*}$
::40磅0.454公斤1磅=18.144公斤。

Let's call the dose needed x , and set up the proportion.
::让我们调用需要的剂量 x, 并设定比例。

$\begin{aligned} \frac{x}{18.144} & = \frac{20}{1} \\ x & = 18.144 \cdot 20 = 362.874 \end{aligned}$
$\begin{align*}\frac{x}{18.144}&=\frac{20}{1}\\ x&=18.144 \cdot 20 = 362.874\end{align*}$ This child needs about 363 mg of medicine per dose.
::x.18.144=201x=18.14420=362.874 这个孩子每剂量需要363毫克药物

WARNING
::警告

Cross-multiplication CAN ONLY BE USED when solving a proportion. It cannot be used when performing operations— addition , subtraction , multiplication, or division—with rational expressions.
::在解析某一比例时只能使用交叉乘法,在以合理表达方式进行增加、减法、乘法或除法等操作时不能使用。

Feature: A Spoonful of Sugar Helps the Medicine Go Down
::特色:一勺糖帮助药物下下沉

by Meredith Beaton
::梅雷迪思·贝顿(签名)

We have often heard that children are not just little adults, but nowhere is that more true than in medicine. Medication errors in children are one of the most common problems in medical mismanagement. Children are not only smaller than adults, they also have unique metabolisms that need to be taken into account.
::我们经常听到,儿童不仅仅是小成年人,但最真实的莫过于医药。儿童药品错误是医疗管理不善中最常见的问题之一。儿童不仅小于成年人,而且还有独特的新陈代谢,需要加以考虑。

Why It Matters
::为何重要

Clearly, children are much smaller than most adults, so they would obviously need a smaller amount of medication. But children also have different rates of metabolism than adults, often much slower since their digestive systems are less mature than those of adults. This means it is easier for children to build up toxic levels of drugs in their bodies. Similarly, because their kidneys are immature, it can take longer for children to excrete drugs, also leading to toxic drug levels more easily than in adults.
::显然,儿童比大多数成年人要小得多,因此他们显然需要较少的药物。但是,儿童的新陈代谢率也与成年人不同,因为他们的消化系统不如成年人成熟,往往要慢得多。这意味着儿童更容易在身体中积累有毒水平的药物。同样,由于他们的肾脏不成熟,儿童需要更长的时间来排出药物,也比成年人更容易导致有毒水平的药物。

When prescribing medications, health-care providers determine the quantity of medication to give based on the weight (in kilograms) of the child. Sometimes this becomes even more complicated because liquid medications are prescribed in multidose packages. (Think about cough syrup—it is sold in a bottle that can be used for multiple doses.) When administering medication in the hospital, a nurse uses cross-multiplication to determine how much medication to dispense from the package:
::在开药时,保健提供者根据儿童体重(公斤)确定药品的给药数量,有时由于液体药物是在多剂量包中开的处方,这就变得更加复杂。 (关于咳嗽糖浆-它出售在瓶子里,可以用于多种剂量。 )在医院服药时,护士使用交叉倍数来确定从包中取出多少药物:

$\begin{aligned} \frac{Dose Desired}{Dose in the Package} = \frac{Quantity in mL to Dispense}{Volume to be Administered} . \end{aligned}$
$\begin{align*}\frac{\text{Dose Desired}}{\text{Dose in the Package}}=\frac{\text{Quantity in mL to Dispense}}{\text{Volume to be Administered}}.\end{align*}$
::包装数量(毫升)中所含的剂量渴望剂量,以去除要管理的排泄量。

by gslishman illustrates weight-based dosing of pediatric medicines like acetaminophen and ibuprofen via syringe.
::例如乙氨基苯和ibuprofen等儿科药物的重量剂量。





Summary
::摘要

To solve a rational equation by cross-multiplication, multiply the numerator of one fraction with the denominator of the other fraction, and set the expressions equal to each other. Solve the equation and check for extraneous solutions.
::要通过交叉乘法解析一个理性方程,将一个分数的分子与另一个分数的分母相乘,并将表达式对等。解析方程并检查不相干的解决办法。

Review
::回顾

1. Is $\begin{align*}x=\text{-}2\end{align*}$ a solution to $\begin{align*}\frac{x-1}{x-4}=\frac{x^2-1}{x+4}\end{align*}$ ?
::1. x=-2 是X-1x-4=x2-1x+4的解决方案吗?

Solve the following rational equations:
::解决以下合理方程式:

2. $\begin{align*}\frac{2x}{x+3}=\frac{8}{x}\end{align*}$
::2. 2xx+3=8x

3. $\begin{align*}\frac{4}{x+1}=\frac{x+2}{3}\end{align*}$
::3. 4x+1=x+23

4. $\begin{align*}\frac{x^2}{x+2}=\frac{x+3}{2}\end{align*}$
::4. x2x+2=x+32

5. $\begin{align*}\frac{3x}{2x-1}=\frac{2x+1}{x}\end{align*}$
::5. 3x2x-1=2x+1x

6. $\begin{align*}\frac{x+2}{x-3}=\frac{x}{3x-2}\end{align*}$
::6. x+2x-3=x3x-2

7. $\begin{align*}\frac{x+3}{-3}=\frac{2x+6}{x-3}\end{align*}$
::7. x+3-3=2x+6x-3

8. $\begin{align*}\frac{2x+5}{x-1}=\frac{2}{x-4}\end{align*}$
::8. 2x+5x-1=2x-4

9. $\begin{align*}\frac{6x-1}{4x^2}=\frac{3}{2x+5}\end{align*}$
::9. 6x-14x2=32x+5

10. $\begin{align*}\frac{5x^2+1}{10}=\frac{x^3-8}{2x}\end{align*}$
::10. 5x2+110=x3-82x

11. $\begin{align*}\frac{x^2-4}{x+4}=\frac{2x-1}{3}\end{align*}$
::11. x2-4x+4=2x-13

Explore More
::探索更多

1. Determine the values of a that make each statement true. If there no values, write none.
::1. 确定使每个语句真实的数值,如果没有数值,写无。

a. $\begin{align*}\frac{1}{x-a}=\frac{x}{x+a}\end{align*}$ , such that there is no solution.
::a. 1x-a=xx+a,没有解决办法。

b. $\begin{align*}\frac{1}{x-a}=\frac{x}{x-a}\end{align*}$ , such that there is no solution.
::b. 1x-a=xx-a,因此没有解决办法。

2. A scale model of a race car is in the ratio of 1: x to the real race car. The length of the model is $\begin{align*}2x-21\end{align*}$ units, and the length of the real race car is $\begin{align*}x^2\end{align*}$ units. What is the value of x ?
::2. 赛车的规模型号为1:x与实际赛车的比率为1:x,型号为2x-21,实际赛车的长度为x2。x值是多少?

3. Ramona reads 6 more pages than Jessica every hour. Ramona can read 90 pages when Jessica reads 60 pages. How many pages, x , can Jessica read per hour?
::3. Ramona每小时读6页,比Jessica多读6页,Ramona每小时读90页,Jessica每小时读60页,Jessica每小时能读多少页?

4. Sterling silver is a metal alloy composed of 92.5% silver and 7.5% copper by weight. Jewelry silver is composed of 80% silver and 20% copper by weight. How many ounces of pure silver, x , should be added to 15 ounces of jewelry silver to make sterling silver?
::4. 硬银是一种金属合金,由92.5%的银和7.5%的铜(重量)组成;珠宝银由80%的银和20%的铜(重量)组成;纯银x应该加到15盎司的首饰银中,以制成硬银?

5. The coffee Jim likes is composed of 98.5% water and 1.5% cocoa. The coffee John likes is composed of 97% water and 3% cocoa. How many ounces of water, x, should be added to 20 ounces of coffee that John likes to make the coffee that Jim likes?
::5. 吉姆喜欢的咖啡由98.5%的水和1.5%的可可组成,约翰喜欢的咖啡由97%的水和3%的可可组成,约翰喜欢的咖啡应加多少盎司的水x,加到20盎司的咖啡中,约翰喜欢的咖啡是吉姆喜欢的咖啡?

6. On an architect's scale drawing of a home, $\begin{align*}\frac{1}{4}\end{align*}$ inch represents 16 feet. What length does a measure of 2 and $\begin{align*}\frac{1}{2}\end{align*}$ inches?
::6. 在建筑师对房屋进行规模绘制时,14英寸代表16英尺,2英寸和12英寸的尺度是多少?

7. On a map, each inch represents 8.5 miles. What is the distance represented by 3.5 inches on the map?
::7. 在地图上,每英寸代表8.5英里,地图上3.5英寸代表的距离是多少?

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section.
::尝试一下这种互动关系,加强本节所探讨的概念。

章节大纲

Solving Rational Equations by Cross-Multiplication ::通过交叉重复解决理性等号

Cross-Multiplication ::交叉重复

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

WARNING ::警告

Feature: A Spoonful of Sugar Helps the Medicine Go Down ::特色:一勺糖帮助药物下下沉

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

Solving Rational Equations by Cross-Multiplication
::通过交叉重复解决理性等号

Cross-Multiplication
::交叉重复

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

WARNING
::警告

Feature: A Spoonful of Sugar Helps the Medicine Go Down
::特色:一勺糖帮助药物下下沉

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX