章节大纲

  • Applications of Linear Systems
    ::线性系统的应用

    In this section, we’ll see how consistent , inconsistent and dependent systems might arise in applications.
    ::在本节中,我们将看到在应用中会产生多么一致、前后不一致和依赖性的系统。

    Real-World Application: Yearly Membership 
    ::实际世界申请:年度会员资格

    The movie rental store CineStar offers customers two choices. Customers can pay a yearly membership of $45 and then rent each movie for $2 or they can choose not to pay the membership fee and rent each movie for $3.50. How many movies would you have to rent before the membership becomes the cheaper option?
    ::电影租借店CineStar为客户提供了两种选择。 客户每年可以支付45美元的会员费,然后每部电影租2美元,或者他们可以选择不支付会员费,每部电影租3.50美元。 在会员成为更廉价的选择之前,你还要租多少部电影?

    Let’s translate this problem into algebra. Since there are two different options to consider, we can write two different equations and form a system.
    ::让我们将这个问题转换为代数。 由于有两种不同的选择需要考虑,我们可以写两个不同的方程式并形成一个系统。

    The choices are “membership” and “no membership.” We’ll call the number of movies you rent x and the total cost of renting movies for a year y .
    ::这些选择是“会员 ” 和“没有会员 ” 。 我们将调用您租用的电影数量 x 和 租赁一年的电影总成本 y 。

      flat fee rental fee total
    membership $45 2 x y = 45 + 2 x
    no membership $0 3.50 x y = 3.5 x

    The flat fee is the dollar amount you pay per year and the rental fee is the dollar amount you pay when you rent a movie. For the membership option the rental fee is 2 x , since you would pay $2 for each movie you rented; for the no membership option the rental fee is 3.50 x , since you would pay $3.50 for each movie you rented.
    ::平价费用是您每年支付的美元,租金是您租电影时支付的美元。 对于会员选择,租赁费是2x,因为您每租一部电影都要支付2美元;对于没有会员选择,租赁费是3.50x,因为您每租一部电影都要支付3.50美元。

    Our system of equations is:
    ::我们的方程式系统是:

    y = 45 + 2 x y = 3.50 x

    ::y=45+2xy=3.50x

    Here’s a graph of the system:
    ::以下是系统图:

    Now we need to find the exact intersection point. Since each equation is already solved for y , we can easily solve the system with substitution. Substitute the second equation into the first one:
    ::现在我们需要找到确切的交叉点。 因为每个方程式已经为y解决了, 我们很容易用替代解决系统。 将第二个方程式替换为第一个方程式 :

    y = 45 + 2 x   ⇒ 3.50 x = 45 + 2 x ⇒ 1.50 x = 45 ⇒ x = 30   movies y = 3.50 x

    ::y=45+2x = 3.50x = 45+2x = 1.50x = 45x = 30 phensy= 3.50x

    You would have to rent 30 movies per year before the membership becomes the better option.
    ::在会员成为更好的选择之前,你必须每年租30部电影。

    This example shows a real situation where a consistent system of equations is useful in finding a solution. Remember that for a consistent system, the lines that make up the system intersect at single point. In other words, the lines are not parallel or the slopes are different.
    ::这个示例显示了一种真实情况, 即一个一致的方程式系统有助于找到解决方案。 记住, 对于一个一致的系统, 构成系统的线条会在单一点交叉。 换句话说, 线条不平行, 或者斜坡不同 。

    In this case, the slopes of the lines represent the price of a rental per movie. The lines cross because the price of rental per movie is different for the two options in the problem
    ::在这种情况下,线条的斜坡代表每部电影的租金价格。 线条交叉是因为每部电影的租金价格与问题的两个选项不同。

    Now let’s look at a situation where the system is inconsistent. From the previous explanation, we can conclude that the lines will not intersect if the slopes are the same (and the y − intercept is different). Let’s change the previous problem so that this is the case.
    ::现在让我们看看系统不一致的情况。 从先前的解释看,我们可以得出结论,如果斜坡相同(而y-interview也不同 ) , 线条就不会交叉。 让我们改变前一个问题,这样就可以了。

    Real-World Application: Comparing Options 
    ::真实世界应用程序:比较选项

    Two movie rental stores are in competition. Movie House charges an annual membership of $30 and charges $3 per movie rental. Flicks for Cheap charges an annual membership of $15 and charges $3 per movie rental. After how many movie rentals would Movie House become the better option?
    ::电影院每年收取30美元的会员费,每部电影租金收取3美元。 廉价的Flicks公司每年收取15美元的会员费,每部电影租金收取3美元。 电影院在有多少电影院租金之后会成为更好的选择?

    It should already be clear to see that Movie House will never become the better option, since its membership is more expensive and it charges the same amount per movie as Flicks for Cheap.
    ::看到电影院永远不会成为更好的选择应该已经很清楚了, 因为电影院的会员更贵, 每部电影的收费与Flicks的廉价电影一样高。

    The lines on a graph that describe each option have different y − intercepts—namely 30 for Movie House and 15 for Flicks for Cheap—but the same : 3 dollars per movie. This means that the lines are parallel and so the system is inconsistent.
    ::描述每个选项的图表上的线条有不同的 Y - intercuts- 即电影院30 和 Flicks 15 和 便宜。 但同样: 每部电影3美元。 这意味着线条是平行的,因此系统是不一致的。

    Now let’s see how this works algebraically. Once again, we’ll call the number of movies you rent x and the total cost of renting movies for a year y .
    ::现在让我们看看这在代数方面是如何运作的。 再一次,我们将调用您租用的电影数量 x 和 一年y 中租赁电影的总成本 。

      flat fee rental fee total
    Movie House $30 3 x y = 30 + 3 x
    Flicks for Cheap $15 3 x y = 15 + 3 x

    The system of equations that describes this problem is:
    ::描述这一问题的方程式系统是:

    y = 30 + 3 x y = 15 + 3 x

    ::y=30+3xy=15+3x

    Let’s solve this system by substituting the second equation into the first equation:
    ::让我们用将第二个方程式替换为第一个方程式来解决这个系统:

    y = 30 + 3 x ⇒ 15 + 3 x = 30 + 3 x ⇒ 15 = 30 This statement is always false. y = 15 + 3 x

    ::y=30+3x%15+3x=30+3x@15=30 此语句总是假的 y=15+3x

    This means that the system is inconsistent.
    ::这就是说,这个系统是不一致的。

    Real-World Application: Price of Fruit 
    ::现实世界应用:水果价格

    Peter buys two apples and three bananas for $4. Nadia buys four apples and six bananas for $8 from the same store. How much does one banana and one apple costs?
    ::彼得买两个苹果和三个香蕉,四美元。 Nadia从同一个商店买四个苹果和六个香蕉,八美元。 一个香蕉和一个苹果要多少钱?

    We must write two equations: one for Peter’s purchase and one for Nadia’s purchase.
    ::我们必须写两个方程式:一个是Peter的购买,另一个是Nadia的购买。

    Let’s say a is the cost of one apple and b is the cost of one banana.
    ::假设一个苹果的成本, b 是香蕉的成本。

      cost of apples cost of bananas total cost
    Peter 2 a 3 b 2 a + 3 b = 4
    Nadia 4 a 6 b 4 a + 6 b = 8

    The system of equations that describes this problem is:
    ::描述这一问题的方程式系统是:

    2 a + 3 b = 4 4 a + 6 b = 8

    ::2a+3b=44a+6b=8

    Let’s solve this system by multiplying the first equation by -2 and adding the two equations:
    ::让我们将第一个方程式乘以 - 2 并添加两个方程式来解决这个问题:

    − 2 ( 2 a + 3 b = 4 ) − 4 a − 6 b = − 8   ⇒   4 a + 6 b = 8 4 a + 6 b = 8 _       0 + 0 = 0

    ::- (2a+3b=4) -4a -6b=8 4a+6b=84a+6b=8_0+0=0)

    This statement is always true. This means that the system is dependent.
    ::这个说法总是对的,这意味着这个系统是依附的。

    Looking at the problem again, we can see that we were given exactly the same information in both statements. If Peter buys two apples and three bananas for $4, it makes sense that if Nadia buys twice as many apples (four apples) and twice as many bananas (six bananas) she will pay twice the price ($8). Since the second equation doesn’t give us any new information, it doesn’t make it possible to find out the price of each fruit.
    ::再看这个问题,我们可以看到,我们在两个陈述中都得到了完全相同的信息。 如果彼得以4美元购买两个苹果和三个香蕉,那么如果纳迪亚购买苹果(四个苹果)两倍和香蕉(六个香蕉)两倍于苹果(六个香蕉 ) , 那她就会付出两倍于价格(8美元 ) 。 由于第二个方程式没有给我们任何新信息,因此不可能找到每个水果的价格。

    Example
    ::示例示例示例示例

    Example 1
    ::例1

    A baker sells plain cakes for $7 and decorated cakes for $11. On a busy Saturday the baker started with 120 cakes, and sold all but three. His takings for the day were $991. How many plain cakes did he sell that day, and how many were decorated before they were sold?
    ::一个面包师卖普通蛋糕7美元,装饰蛋糕11美元。在一个忙碌的星期六,面包师先卖了120个蛋糕,除了3个以外,都卖掉了。他今天的收银款是991美元。当日他卖了多少个普通蛋糕,在出售之前还装饰了多少个?

      plain cakes decorated cakes total
    Cakes sold p d 120 − 3 = 117
    Cost of cakes 7p 11d $991

    The system of equations that describes this problem is:
    ::描述这一问题的方程式系统是:

    p + d = 117 7 p + 11 d = 991

    ::p+d=11777p+11d=991

    Let’s solve this system by substituting the second equation into the first equation:
    ::让我们用将第二个方程式替换为第一个方程式来解决这个系统:

    p + d = 117 ⇒ p = 117 − d

    ::p+d= 117_p= 117_d

    7 p + 11 d = 991 ⇒ 7 ( 117 − d ) + 11 d = 991 ⇒ 819 − 7 d + 11 d = 991 ⇒ 819 + 4 d = 991 ⇒ 4 d = 172 ⇒ d = 43

    ::7p+11d=9917(117-d)+11d=991819-7d+11d=991819+4d=991819+4d=9914d=1724d=43

    We can substitute d into the first equation to solve for p .
    ::我们可以用第一个方程式代替p

    p = 117 − d = 117 − ( 43 ) = 74
    ::p=117-d=117-(43)=74

    The baker sold 74 plain cakes and 43 decorated cakes.
    ::面包师卖了74个普通蛋糕和43个装饰蛋糕。

    Review 
    ::回顾

    1. Twice John’s age plus five times Claire’s age is 204. Nine times John’s age minus three times Claire’s age is also 204. How old are John and Claire?
      ::约翰的两岁加克莱尔的五倍是204岁。 约翰的九倍减去克莱尔的三倍是204岁。 约翰和克莱尔几岁?
    2. Juan is considering two cell phone plans. The first company charges $120 for the phone and $30 per month for the calling plan that Juan wants. The second company charges $40 for the same phone but charges $45 per month for the calling plan that Juan wants. After how many months would the total cost of the two plans be the same?
      ::Juan正在考虑两个手机计划。 第一家公司为电话收费120美元,为Juan想要的电话计划每月收费30美元。 第二家公司为同一部电话收费40美元,但为Juan想要的电话计划每月收费45美元。 在这两个计划的总成本是多少个月之后?
    3. Jamal placed two orders with an internet clothing store. The first order was for 13 ties and 4 pairs of suspenders, and totaled $487. The second order was for 6 ties and 2 pairs of suspenders, and totaled $232. The bill does not list the per-item price, but all ties have the same price and all suspenders have the same price. What is the cost of one tie and of one pair of suspenders?
      ::Jamal向一家互联网服装店发出了两项订单。 第一项订单是13条领带和4双吊带,总额为487美元。 第二项订单是6条领带和2双吊带,总额为232美元。 账单没有列出每件物品的价格,但所有领带的价格都一样,所有吊带的价格都一样。 一条领带和一双吊带的成本是多少?
    4. An airplane took four hours to fly 2400 miles in the direction of the jet-stream. The return trip against the jet-stream took five hours. What were the airplane’s speed in still air and the jet-stream's speed?
      ::一架飞机用4小时朝喷气流方向飞行24英里。 对喷气流的回程历时5小时。 飞机在空气中的速度和喷气流的速度是多少?

    For questions 5-7, a movie theater charges $4.50 for children and $8.00 for adults.
    ::对于问题5-7,一个电影院收取儿童4.50美元和成人8.00美元的费用。

    1. On a certain day, 1200 people enter the theater and $8375 is collected. How many children and how many adults attended?
      ::在某一天,1200人进入剧院,8375美元被收买,有多少儿童和多少成年人参加?
    2. The next day, the manager announces that she wants to see them take in $10000 in tickets. If there are 240 seats in the house and only five movie showings planned that day, is it possible to meet that goal?
      ::第二天,经理宣布,她想看到他们以1 000美元的票价买票。 如果家里有240个座位,而且当天只计划了5个电影放映,那么能否达到这个目标?
    3. At the same theater, a 16-ounce soda costs $3 and a 32-ounce soda costs $5. If the theater sells 12,480 ounces of soda for $2100, how many people bought soda? ( Note: Be careful in setting up this problem!)
      ::在同一剧院,一辆16盎司苏打水需要3美元,一辆32盎司苏打水需要5美元。 如果剧院出售12,480盎司苏打水需要2100美元,有多少人买苏打水? (注:小心设置问题! )

    For questions 8-10, consider the situation: Nadia told Peter that she went to the farmer’s market and bought two apples and one banana, and that it cost her $2.50. She thought that Peter might like some fruit, so she went back to the seller and bought four more apples and two more bananas. Peter thanked Nadia, but told her that he did not like bananas, so he would only pay her for four apples. Nadia told him that the second time she paid $6.00 for the fruit.
    ::对于问题8-10,请考虑一下情况:Nadia告诉Peter,她去了农贸市场,买了两块苹果和一个香蕉,花了她2.50美元。 她认为Peter可能喜欢一些水果,因此她回到卖家那里,又买了四块苹果和两块香蕉。 Peter感谢Nadia,但告诉她他不喜欢香蕉,所以他只付她四块苹果。 Nadia告诉他,她第二次为水果支付了6美元。

    1. What did Peter find when he tried to figure out the price of four apples?
      ::Peter想弄清楚4个苹果的价格时 发现了什么?
    2. Nadia then told Peter she had made a mistake, and she actually paid $5.00 on her second trip. Now what answer did Peter get when he tried to figure out how much to pay her?
      ::Nadia后来告诉Peter她犯了一个错误,她第二次旅行支付了5美元。当Peter试图弄清楚要付她多少钱时,他得到了什么答案?
    3. Alicia then showed up and told them she had just bought 3 apples and 2 bananas from the same seller for $4.25. Now how much should Peter pay Nadia for four apples?
      ::Alicia后来出现,告诉他们她刚从同一家卖家那里买了3个苹果和2个香蕉,价格是4.25美元。 现在Peter应该为4个苹果付给Nadia多少钱?

    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 。