课程： 5.6 线性赤道的书写和解决系统-interactive

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线性赤道的书写和溶解系统

Solar Energy
::太阳能太阳能

In this section, you will explore how to write and solve systems of equations from real-world situations. When you encounter systems of equations in the real world, you will not have the equations. The ability to represent situations with equations is extremely important. For example, a renewable energy analyst needs to be able to compare the various forms of renewable energy in order to determine which ones are the most efficient in a variety of situations.
::在本节中,您将探索如何从现实世界中写入和解析方程式系统。当遇到现实世界中的方程式系统时, 您将没有方程式。以方程式代表各种方程式的能力是极为重要的。例如, 可再生能源分析师需要能够比较各种形式的可再生能源, 以便确定在各种情况下哪些是最有效的。

Take a look at a real situation which every homeowner faces − saving on energy costs. Home owners are looking at solar panels as a source of renewable energy to save money. However, before making a purchase, it is important to understand how much money they will save and how long the homeowner will have to have the solar panels before they pay themselves off. There are many options for renewable energy, and the ability to perform an analysis will result in saving the most money.
::审视一下每个房主面临的真实状况 — — 节约能源成本。房主将太阳能电池板视为节约资金的可再生能源来源。然而,在购买之前,重要的是要了解他们将节省多少资金,以及房主需要多久才能得到太阳能电池板才能偿还。可再生能源有许多选择,而进行分析的能力将导致节约大部分资金。

Solar Energy Continued
::太阳能继续

Identify the Problem or Question
::查明问题或问题

You are considering getting solar panels for the roof of your house. You want to know if solar panels are a better option than what you have now.
::你正在考虑为房子的屋顶买太阳能电池板。你想知道太阳能电池板是否比现在更好的选择。

Gather Data and Research
::收集数据和研究

The solar panels and installation will cost you $17,000. The inspector tells you that your new monthly bill is estimated to be $55 per month. Your current bill is $124 per month on average.
::太阳能电池板和安装费用为17,000美元。检查员告诉你,你每月的新账单估计为每月55美元。你目前的账单平均为每月124美元。

Analyze the Data
::分析数据

INTERACTIVE

Solar Panels

Move the red dot to see how the cost should increase with each month
::将红色点移动到每个月的费用增加情况

To determine how long it will take the solar panels to pay themselves off we must write a system of equations which models this situation. First, look at the two choices: the energy cost of solar panels and the energy cost without solar panels. You need to write equations that will show how the overall cost changes ( $\begin{aligned} y \end{aligned}$ ) by the month ( $\begin{aligned} x \end{aligned}$ ).
::为了确定太阳能电池板要花多久才能偿还费用,我们必须写出一个模拟这种情况的方程式系统。首先,看看两种选择:太阳能电池板的能源成本和没有太阳能电池板的能源成本。你需要写出能够显示每月(十)总成本变化的方程式。

Without solar panels, y ou pay an average of $124 per month. Your total cost will be $124 multiplied by the number of months. This can be modeled with the equation $\begin{aligned} y = 124 x . \end{aligned}$
::没有太阳能电池板,您每月平均支付124美元。您的总费用为124美元乘月数。这可以用y=124x的方程式来模拟。

With solar panels, you pay an average of $55 per month and have to pay an upfront cost of $17,000. Your total cost will be $55 multiplied by the number of months plus $17,000. This can be modeled with the equation : $\begin{aligned} y = 55 x + 17, 000 \end{aligned}$ .
::太阳能电池板每月平均支付55美元,预付费用为17 000美元,总费用为55美元乘月数加17 000美元。这可以用公式y=55x+17 000来模拟。

Since both of these equations are in $\begin{aligned} y - \end{aligned}$ intercept form and knowing more than just the solution may help with your analysis, graphing may be the best approach.
::因为这两个方程式都以y-inter-interform形式存在, 不仅知道解决办法可能有助于你的分析, 图形化可能是最好的方法。

INTERACTIVE

Find the Intersection

Input equations into the boxes and click on their intersection if it exists to see its coordinates.
::输入方程式进入框中, 并点击它们的交叉点, 如果它存在可以看到它的坐标。

Drag and zoom to find the intersection.
::拖曳和缩放以找到交叉点。

If lost, press the reset button to move back to the origin.
::如果丢失, 请按重设按钮返回来源。

Discussion Question
::讨论问题

Discuss with a classmate the information can you take from this graph other than the solution? Were your answers the same? Do you agree with your partner? Why or why not?
::与同学讨论您可从此图中除解答之外获取的信息。您的答案是否相同 ? 您是否同意您的伴侣 ? 为什么或为什么不呢 ?

Take Action
::采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动采取行动

Now that you know the solar panels will pay themselves off in 246 months, or 20 1/2 years, what would you do?
::现在你知道太阳能电池板在246个月或20个半年后将会自收其利你会怎么做?

Garden Design
::花园设计

Cecelia is planning a community garden. She plans to buy 150 feet of fencing to keep animals out. If she wants the length to be 16 less than the width, what will the length and width of the garden be?
::Cecelia正在规划一个社区花园,她计划购买150英尺的栅栏,以阻止动物进入。如果她想要比宽度少16英尺,那么花园的长度和宽度是多少?

You have explored problems similar to this with . However, another approach is to use a system of equations. This approach will be even more important when faced with a problem that involves more than two variables. An example would be having to solve for 3 variables because you have a 3-dimensional object that has length, width, and height.
::您已经与 . 探讨过类似的问题。但是, 另一种方法是使用方程系统。当遇到一个涉及两个变量以上的问题时, 这种方法将更加重要。一个例子是必须解决三个变量, 因为您有一个三维对象, 其长度、宽度和高度都具有长度、宽度和高度。

First, start with the fence. Begin by writing two equations involving the two variables length and width. You know the perimeter is 150 feet and the formula for perimeter is $\begin{aligned} P = 2 L + 2 W \end{aligned}$ . U se this to write the first equation:
::首先,从栅栏开始。开始写两个方程式, 涉及两个变量的长度和宽度。你知道周界是150英尺, 周界的公式是P=2L+2W。用这个公式来写第一个方程式 :

\begin{aligned} 150 = 2 W + 2 L \end{aligned}

::150=2W+2L

Additionally, it is known that the length is 16 less than the width. U se this to write the second equation.
::此外,已知长度比宽度低16。用此写第二个方程。

Length: $\begin{aligned} L \end{aligned}$
::长度:L

is: $\begin{aligned} = \end{aligned}$
::是: =

16 less than : $\begin{aligned} - 16 \end{aligned}$
::16小于: -16

the width: $\begin{aligned} W \end{aligned}$
::宽度: W

\begin{aligned} L = W - 16 \end{aligned}

::L=W-166 L=W-16

Now that you have two equations with two variables you can solve them. Since $\begin{aligned} L \end{aligned}$ is already solved for and you do not need a graph to consider other possibilities, substitution is the best approach.
::现在您有了两个方程式, 有两个变量可以解答它们。因为 L 已经解答了, 而且您不需要图表来考虑其他的可能性, 替换是最好的方法。

\begin{aligned} L & = W - 16 \\ 150 & = 2 W + 2 L \end{aligned}

::L=W-16150=2W+2L

\begin{aligned} 150 & = 2 W + 2 (W - 16) \\ 150 & = 2 W + 2 W - 32 \\ 150 & = 4 W - 32 \\ 182 & = 4 W \\ W & = 45 \frac{1}{2} \end{aligned}

::150=2W+2(W-16)150=2W+2W-32150=4W-322182=4WW=4512

Now that you know $\begin{aligned} W, \end{aligned}$ plug it into one of the original equations to determine $\begin{aligned} L . \end{aligned}$ Now substitute it into the equation $\begin{aligned} L = W - 16. \end{aligned}$
::现在你知道W了, 把它插进一个原始方程来决定L。现在把它换成L=W -16的方程。

\begin{aligned} L & = W - 16 \\ L & = (45 \frac{1}{2}) - 16 \\ L & = 29 \frac{1}{2} \end{aligned}

::LL=W-16L=(4512)-16L=2912

The dimensions of the garden are 29 ½ ft by 45 ½ ft.
::花园的尺寸是29个半英尺乘45个半英尺。

Taking a Trip
::旅行旅行

Example
::示例示例示例示例

Brandon is taking a trip from Boston to San Diego. The plane traveled 550 mph on the way to San Diego and 605 mph on the way back to Boston. The trip took 30 minutes more to get to San Diego than it did coming back to Boston. Assuming that there were no stops and that the pilot traveled the same path both ways, what is the distance between Boston and San Diego?
::Brandon正从波士顿前往圣地亚哥。飞机在前往圣地亚哥的途中飞行了550英里,在返回波士顿的途中飞行了605英里。前往圣地亚哥比返回波士顿要花30分钟多。假设没有停停,飞行员也走同一条路,波士顿和圣地亚哥之间的距离是多少?

Motion problems are common on standardized tests. They present two situations with moving objects and you must determine a missing piece of information based on the relationship between speed , distance, and time. These problems can be tricky but you can set up a table to help.
::运动问题在标准测试中是常见的。它们呈现着两个移动对象的情况, 您必须根据速度、距离和时间之间的关系来确定缺少的信息。这些问题可能比较棘手, 但是您可以设置一个表格来提供帮助。

	Distance	Rate	Time
Trip to San Diego	$\begin{aligned} d \end{aligned}$	$\begin{aligned} 550 \end{aligned}$	$\begin{aligned} t + 0.5 \end{aligned}$
Trip to Boston	$\begin{aligned} d \end{aligned}$	$\begin{aligned} 605 \end{aligned}$	$\begin{aligned} t \end{aligned}$

The distance from Boston to San Diego is the same as the distance from San Diego to Boston so the variable $\begin{aligned} d \end{aligned}$ was used for both. The trip to San Diego took 30 minutes more than the trip back to Boston, so the time to Boston is written as $\begin{aligned} t \end{aligned}$ and the time to San Diego is written as $\begin{aligned} t + 0.5. \end{aligned}$ When writing this equation write 30 minutes in terms of hours (0.5) because the rate is in miles per hour. Now, use the relationship $\begin{aligned} distance = rate \cdot time \end{aligned}$ to write the two equations.
::从波士顿到圣地亚哥的距离与从圣地亚哥到波士顿的距离相同,因此变数 d 用于两者。圣地亚哥之行比回波士顿的行程花费了30分钟,所以去波士顿的时间写为t,去波士顿的时间写为t+0.5。写这个方程式时用小时( 0. 5) 写为30分钟, 因为速率是每小时的英里。现在, 使用关系距离时间来写两个方程。

S ince the variable $\begin{aligned} d \end{aligned}$ is solved for in both equations and you do not need a graph, using substitution will likely be the most efficient method .
::由于变量d在两个方程式中均被解答,而您不需要图表,使用替代可能是最有效的方法。

You can substitute 5 in for the variable $\begin{aligned} t \end{aligned}$ in the equation $\begin{aligned} d = 605 t \end{aligned}$ to find the distance.
::您可以替换公式 d=605t 中的变量 t 5, 以找到距离。

\begin{aligned} d & = 605 (5) \\ d & = 3, 025 \end{aligned}

::d=605(5)d=3 025

The distance from Boston to San Diego based on the information gathered by the trip is 3,025 miles.
::从波士顿到圣地亚哥的距离根据这次旅行收集的信息距离是3 025英里

Summary

The solution on the graph of a system of equations is the point where the two lines intersect.
::方程系统图上的解决方案是两条线相交的点。

The solution to a system of equations is the point at which the two equations have the same input value and the same output.
::方程式系统的解决方案是两个方程式具有相同输入值和相同输出值的点。

To solve a system of equations use the substitution method, elimination method, or graph to find the point of intersection.
::要解决方程式系统使用替代方法,消除方法,或图形找到交叉点。

章节大纲

常规

线性赤道的书写和溶解系统

Solar Energy ::太阳能太阳能

Solar Energy Continued ::太阳能继续

Garden Design ::花园设计

Taking a Trip ::旅行旅行

Solar Energy
::太阳能太阳能

Solar Energy Continued
::太阳能继续

Garden Design
::花园设计

Taking a Trip
::旅行旅行