课程： 7.4 增加或减法的线性系统

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选择节增加或减法的线性系统

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增加或减法的线性系统
Solving Linear Systems by Elimination
::通过消除解决线性系统

You can use simple addition and subtraction to simplify a system of equations to a single equation involving a single variable . Because we go from two unknowns ( $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ ) to a single unknown (either $\begin{aligned} x \end{aligned}$ or $\begin{aligned} y \end{aligned}$ ), this method is often referred to as solving by elimination . We eliminate one variable in order to make our equations solvable! To illustrate this idea, let’s look at the simple example of buying apples and bananas.
::您可以使用简单的加法和减法来简化单方程系统, 以简化单一变量的单方程。由于我们从两个未知数( x 和 y ) 到一个未知数( x 或 y ) , 这种方法通常被指为以除去方式解决。我们删除一个变量, 以使我们的方程可以溶解。为了说明这一点, 让我们看看购买苹果和香蕉的简单例子。

If one apple plus one banana costs $1.25 and one apple plus 2 bananas costs $2.00, how much does one banana cost? One apple?
::如果一个苹果加一个香蕉需要1.25美元, 一个苹果加2个香蕉需要2.00美元, 一个香蕉要多少钱?

It shouldn’t take too long to discover that each banana costs $0.75. After all, the second purchase just contains 1 more banana than the first, and costs $0.75 more, so that one banana must cost $0.75.
::不久就能发现每只香蕉都要花0.75美元。毕竟,第二次购买的香蕉比第一次购买的香蕉多出1美元,费用为0.75美元,因此,一个香蕉必须花0.75美元。

Here’s what we get when we describe this situation with algebra:
::以下是我们用代数描述这种情况时得到的结果:

$\begin{aligned} a + b & = 1.25 \\ a + 2 b & = 2.00 \end{aligned}$

::a+b=1.25a+2b=2.00

Now we can subtract the number of apples and bananas in the first equation from the number in the second equation, and also subtract the cost in the first equation from the cost in the second equation, to get the difference in cost that corresponds to the difference in items purchased.
::现在,我们可以从第二个方程的数字中减去第一个方程中的苹果和香蕉数量,并从第二个方程中减去第一个方程中的成本,以获得与购买物品差异相对应的成本差异。

$\begin{aligned} (a + 2 b) - (a + b) = 2.00 - 1.25 \to b = 0.75 \end{aligned}$

:a)+2b)-(a+b)=2.00-1.25b=0.75

That gives us the cost of one banana. To find out how much one apple costs, we subtract $0.75 from the total cost of one apple and one banana.
::这让我们损失了一只香蕉的成本。为了了解一个苹果的成本,我们从一个苹果和一个香蕉的总成本中扣除了0.75美元。

$\begin{aligned} a + 0.75 = 1.25 \to a = 1.25 - 0.75 \to a = 0.50 \end{aligned}$

::a+0.75=1.25a=1.25-0.75a=0.50。

So an apple costs 50 cents.
::所以苹果要花50美分

To solve systems using addition and subtraction, we’ll be using exactly this idea – by looking at the sum or difference of the two equations we can determine a value for one of the unknowns.
::为了解决使用增减法的系统, 我们将使用这个想法 — 查看两个方程式的总和或差, 我们可以确定一个未知方程式的值。

Solving Linear Systems Using Addition of Equations
::使用增加等量的溶解线性系统

Often considered the easiest and most powerful method of solving systems of equations, the addition (or elimination) method lets us combine two equations in such a way that the resulting equation has only one variable. We can then use simple algebra to solve for that variable. Then, if we need to, we can substitute the value we get for that variable back into either one of the original equations to solve for the other variable.
::通常会考虑最简单和最有力的解析方程式系统的方法,添加(或消除)方法让我们将两个方程式组合在一起,使得由此产生的方程式只有一个变量。然后我们可以使用简单的代数来解决该变量。然后,如果我们需要,我们可以将变量获得的值替换为用于解决另一个变量的原始方程式之一。

Solve this system by addition:
::除此以外,还解决了这一系统:

$\begin{aligned} 3 x + 2 y & = 11 \\ 5 x - 2 y & = 13 \end{aligned}$

::3x+2y=115x-2y=13

We will add everything on the left of the equals sign from both equations, and this will be equal to the sum of everything on the right:
::我们必在两种方程的等距左边加一切等距,这等于右边每样物的总和。

$\begin{aligned} (3 x + 2 y) + (5 x - 2 y) = 11 + 13 \to 8 x = 24 \to x = 3 \end{aligned}$

:3x+2y)+(5x-2y)=11+13_8x=24x=3

A simpler way to visualize this is to keep the equations as they appear above, and to add them together vertically , going down the columns. However, just like when you add units, tens and hundreds, you MUST be sure to keep the $\begin{aligned} x \end{aligned}$ 's and $\begin{aligned} y \end{aligned}$ 's in their own columns. You may also wish to use terms like " $\begin{aligned} 0 y \end{aligned}$ " as a placeholder!
::更简单的视觉化方法就是保持方程式的上方, 并垂直地将它们相加, 沿着柱子往下移。但是, 就像您添加单位, 数以万计, 数以百计, 您必须把 x 和 y 保存在自己的列内。您也可以使用“ 0y” 这样的词作为占位符 !

$\begin{aligned} 3 x + 2 y = 11 \\ \underline{+ (5 x - 2 y) = 13} \\ 8 x + 0 y = 24 \end{aligned}$

::3x+2y=11+(5x-2y)=13_8x+0y=24

Again we get $\begin{aligned} 8 x = 24 \end{aligned}$ , or $\begin{aligned} x = 3 \end{aligned}$ . To find a value for $\begin{aligned} y \end{aligned}$ , we simply substitute our value for $\begin{aligned} x \end{aligned}$ back in.
::我们再次获得 8x=24, 或 x=3。要找到y 的值, 我们只需将我们的值替换为 x 返回。

Substitute $\begin{aligned} x = 3 \end{aligned}$ into the second equation:
::第二个方程中的替代 x=3 :

$\begin{aligned} 5 \cdot 3 - 2 y & = 13 & s i n c e 5 \times 3 = 15, w e s u b t r a c t 15 f r o m b o t h s i d e s \\ - 2 y & = - 2 & d i v i d e b y - 2 \\ y & = 1 \end{aligned}$

::53--2y=13 自 5×3=15, 我们从两侧减去 15 - 2y=2divide 由 - 2y=1

The reason this method worked is that the $\begin{aligned} y - \end{aligned}$ coefficients of the two equations were opposites of each other: 2 and -2. Because they were opposites, they canceled each other out when we added the two equations together, so our final equation had no $\begin{aligned} y - \end{aligned}$ term in it and we could just solve it for $\begin{aligned} x \end{aligned}$ .
::之所以采用这种方法,是因为两个方程的y-coconvaly 相互对立: 2和 - 2。因为它们是对立的, 当我们把两个方程加在一起时,它们就相互取消, 因此我们最后的方程没有y- term 在其中没有y- term, 我们可以用 x 来解答它。

Solving Linear Systems Using Subtraction of Equations
::使用等量减法解决线性系统

Another, very similar method for solving systems is subtraction. When the $\begin{aligned} x - \end{aligned}$ or $\begin{aligned} y - \end{aligned}$ coefficients in both equations are the same (including the sign) instead of being opposites, you can subtract one equation from the other.
::另一个非常相似的解答系统方法是减法。当两个方程式中的 x- 或 y- 系数相同( 包括符号) 而不是相反时, 您可以从另一个方程式中减去一个方程式。

If you look again at Example 3, you can see that the coefficient for $\begin{aligned} x \end{aligned}$ in both equations is +1. Instead of adding the two equations together to get rid of the $\begin{aligned} y^{'} \end{aligned}$ s, you could have subtracted to get rid of the $\begin{aligned} x^{'} \end{aligned}$ s:
::如果您再看看例3, 您可以看到两个方程式中 x 的系数是+1 。而不是将两个方程式加在一起来除掉 y's, 您本可以减去来除掉 x’s :

$\begin{aligned} (x + y) - (x - y) = 7 - 1.5 \Rightarrow 2 y = 5.5 \Rightarrow y = 2.75 \\ or . . . \\ x + y = 7 \\ \underline{- (x - y) = - 1.5} \\ 0 x + 2 y = 5.5 \end{aligned}$

:x+y)-(x-y)=7-1.5_%2y=5.5y=2.75or... x+y=7-(x-y)\=1.5_0x+2y=5.5

So again we get $\begin{aligned} y = 2.75 \end{aligned}$ , and we can plug that back in to determine $\begin{aligned} x \end{aligned}$ .
::所以,我们再次得到y=2.75, 我们可以插上它来确定 x。

The method of subtraction is just as straightforward as addition, so long as you remember the following:
::减法与加法一样直截了当,只要你记得以下内容:

Always put the equation you are subtracting in " data-term="Parentheses" role="term" tabindex="0"> parentheses , and distribute the negative.
::总是在括号中键入您要减去的方程式,然后将负号分布。

Don’t forget to subtract the numbers on the right-hand side.
::不要忘了减去右手边的数字。

Always remember that subtracting a negative is the same as adding a positive.
::始终记住,减去负数等于加上正数。

Peter examines the coins in the fountain at the mall. He counts 107 coins, all of which are either pennies or nickels. The total value of the coins is $3.47. How many of each coin did he see?
::Peter在商场的喷泉里检查了硬币。他有107个硬币,全部都是硬币或镍。硬币的总价值是3.47美元。他看到每枚硬币中有多少?

We have 2 types of coins, so let’s call the number of pennies $\begin{aligned} x \end{aligned}$ and the number of nickels $\begin{aligned} y \end{aligned}$ . The total value of all the pennies is just $\begin{aligned} x \end{aligned}$ , since they are worth $\begin{aligned} 1 c \end{aligned}$ each. The total value of the nickels is $\begin{aligned} 5 y \end{aligned}$ . We are given two key pieces of information to make our equations: the number of coins and their value in cents.
::我们有两种硬币类型,因此让我们把硬币数量X和镍数量 y 调用。所有硬币的总价值都只是x,因为它们每个值1c。镍的总价值是5y。我们得到两条关键的信息,可以得出我们的方程:硬币数量和以美分计的价值。

$\begin{aligned} # of coins equation : x + y = 107 \\ value equation : x + 5 y = 347 \end{aligned}$

::硬币方程式 #:x+y=107价值方程式:x+5y=347

We’ll jump straight to subtracting the two equations:
::我们直接跳转到减去两个方程式:

$\begin{aligned} x + y = 107 \\ \underline{- (x + 5 y) = - 347} \\ - 4 y = - 240 \\ y = 60 \end{aligned}$

::x+y=107- (x+5y) @%347_- 4y_ 240y=60

Substituting this value back into the first equation:
::将此值重置为第一个方程 :

$\begin{aligned} x + 60 & = 107 & s u b t r a c t 60 f r o m b o t h s i d e s : \\ x & = 47 \end{aligned}$

::x+60=107 从两侧提取60分数:x=47

So Peter saw 47 pennies (worth 47 cents) and 60 nickels (worth $3.00) making a total of $3.47.
::彼得看到47个便士(47美分)和60个镍(3.00美分),

Examples
::实例

Example 1
::例1

Andrew is paddling his canoe down a fast-moving river. Paddling downstream he travels at 7 miles per hour, relative to the river bank. Paddling upstream, he moves slower, traveling at 1.5 miles per hour. If he paddles equally hard in both directions, how fast is the current? How fast would Andrew travel in calm water?
::安德鲁在一条快速移动的河流上划独木舟。与河岸相对,他每小时以7英里的速度在下游行。上游划船慢行,每小时以1.5英里的速度行驶。如果他在双向划船同样艰难,洋流的速度有多快?安德鲁在平静的水中行走多快?

First, we convert our problem into equations. We have two unknowns to solve for, so we’ll call the speed that Andrew paddles at $\begin{aligned} x \end{aligned}$ , and the speed of the river $\begin{aligned} y \end{aligned}$ . When traveling downstream, Andrew speed is boosted by the river current, so his total speed is his paddling speed plus the speed of the river $\begin{aligned} (x + y) \end{aligned}$ . Traveling upstream, the river is working against him, so his total speed is his paddling speed minus the speed of the river $\begin{aligned} (x - y) \end{aligned}$ .
::首先,我们把问题转换成方程式。我们有两个未知因素需要解决,因此我们称其为安德鲁在x的桨速和河速。在下游中,安德鲁的速度被河流加速,所以他的总速度是他的桨速和河速(x+y ) 。在上游,河速对他不利,所以他的总速度是他的桨速减去河速(x-y ) 。

Downstream Equation: $\begin{aligned} x + y = 7 \end{aligned}$
::下游方程式: x+y=7

Upstream Equation: $\begin{aligned} x - y = 1.5 \end{aligned}$
::上流赤道: x-y=1.5

Next we’ll eliminate one of the variables. If you look at the two equations, you can see that the coefficient of $\begin{aligned} y \end{aligned}$ is +1 in the first equation and -1 in the second. Clearly $\begin{aligned} (+ 1) + (- 1) = 0 \end{aligned}$ , so this is the variable we will eliminate. To do this we simply add equation 1 to equation 2. We must be careful to collect like terms , and make sure that everything on the left of the equals sign stays on the left, and everything on the right stays on the right:
::接下来我们将消除其中的一个变量。如果您查看这两个方程, 您可以看到 y 的系数在第一个方程中是+1, 在第二个方程中是-1。显然( +1) +( 1) =0 ) , 所以这就是我们将要消除的变量。要做到这一点, 我们只需在公式 2 中增加等式 1 。我们必须小心收集类似条件, 并确保等式左侧的所有符号都停留在左边, 右侧的一切都停留在右边 :

$\begin{aligned} (x + y) + (x - y) = 7 + 1.5 \Rightarrow 2 x = 8.5 \Rightarrow x = 4.25 \end{aligned}$

:x+y)+(x-y)=7+1.52x=8.5x=4.25

Or, using the column method we used in example 2:
::或者,使用我们在例2中使用的列方法:

$\begin{aligned} x + y = 7 \\ \underline{+ x - y = 1.5} \\ 2 x + 0 y = 8.5 \end{aligned}$

::x+y=7 + x-y=1.5_ 2x+0y=8.5

Again we get $\begin{aligned} 2 x = 8.5 \end{aligned}$ , or $\begin{aligned} x = 4.25 \end{aligned}$ . To find a corresponding value for $\begin{aligned} y \end{aligned}$ , we plug our value for $\begin{aligned} x \end{aligned}$ into either equation and isolate our unknown. In this example, we’ll plug it into the first equation:
::我们再次获得 2x=8.5, 或 x=4. 25。要找到y 的相应值, 我们将 x 的值插入两个方程式中, 并分离未知的方程式。在此示例中, 我们将将其插入第一个方程式 :

$\begin{aligned} 4.25 + y & = 7 & s u b t r a c t 4.25 f r o m b o t h s i d e s : \\ y & = 2.75 \end{aligned}$

::4.25+y=7 双方4.25次数:y=2.75

Andrew paddles at 4.25 miles per hour. The river moves at 2.75 miles per hour.
::安德鲁以每小时4.25英里的速度划桨河流以每小时2.75英里的速度移动

Review
::回顾

Solve the system:
$\begin{aligned} 3 x + 4 y = 2.5 \\ 5 x - 4 y = 25.5 \end{aligned}$

::解决系统:3x+4y=2.55x-4y=25.5

Solve the system:
$\begin{aligned} 2 x + - y = 10 \\ 3 x + y = - 5 \end{aligned}$

::解决系统: 2xy=103x+y5

Solve the system:
$\begin{aligned} 5 x + 7 y = - 31 \\ 5 x - 9 y = 17 \end{aligned}$

::解决系统: 5x+7y315x-9y=17

Solve the system:
$\begin{aligned} 3 y - 4 x = - 33 \\ 5 x - 3 y = 40.5 \end{aligned}$

::解决系统: 3y-4x335x-3y=40.5

Nadia and Peter visit the candy store. Nadia buys three candy bars and four fruit roll-ups for $2.84. Peter also buys three candy bars, but can only afford one additional fruit roll-up. His purchase costs $1.79. What is the cost of a candy bar and a fruit roll-up individually?
::Nadia和Peter参观了糖果店。Nadia以2.84美元购买了3个糖果棒和4个水果卷,Peter也购买了3个糖果条,但只能再买一个水果卷。他的购买成本为1.79美元。一个糖果条和水果卷的成本是多少?

A small plane flies from Los Angeles to Denver with a tail wind (the wind blows in the same direction as the plane) and an air-traffic controller reads its ground-speed (speed measured relative to the ground) at 275 miles per hour. Another, identical plane, moving in the opposite direction has a ground-speed of 227 miles per hour. Assuming both planes are flying with identical air-speeds, calculate the speed of the wind.
::从洛杉矶飞往丹佛的一小架飞机,尾风(风向与飞机的方向相同),空中交通控制器的地面速度(相对于地面的速度)为每小时275英里。另一架方向相反的飞机的地面速度为每小时227英里。假设这两架飞机的飞行速度与飞行速度相同,则计算风速。

An airport taxi firm charges a pick-up fee, plus an additional per-mile fee for any rides taken. If a 12-mile journey costs $14.29 and a 17-mile journey costs $19.91, calculate:

the pick-up fee
::取款费

the per-mile rate
::每英里费率

the cost of a seven mile trip
::7英里旅行费用

::机场出租车公司向机场出租车公司收取提款费,加上任何乘坐的任何旅程的每英里额外收费。如果12英里的行程费用为14.29美元,17英里的行程费用为19.91美元,计算:提款费为7英里的行程费用。

Calls from a call-box are charged per minute at one rate for the first five minutes, then a different rate for each additional minute. If a 7-minute call costs $4.25 and a 12-minute call costs $5.50, find each rate.
::头5分钟每分钟按1美元费率收取呼叫箱电话费,然后每多5分钟按不同的费率收费,如果7分钟的呼叫费为4.25美元,12分钟的呼叫费为5.50美元,则按每个费率收费。

A plumber and a builder were employed to fit a new bath, each working a different number of hours. The plumber earns $35 per hour, and the builder earns $28 per hour. Together they were paid $330.75, but the plumber earned $106.75 more than the builder. How many hours did each work?
::一个水管工和一个建筑工被雇用来适应一个新的浴池,每个水管工的工作时数各异。水管工每小时挣35美元,建筑工每小时挣28美元。他们总共得到330.75美元的工资,但水管工的收入比建筑工多106.75美元。每个工作了多少小时?

Paul has a part time job selling computers at a local electronics store. He earns a fixed hourly wage, but can earn a bonus by selling warranties for the computers he sells. He works 20 hours per week. In his first week, he sold eight warranties and earned $220. In his second week, he managed to sell 13 warranties and earned $280. What is Paul’s hourly rate, and how much extra does he get for selling each warranty?
::保罗在当地一家电子商店有兼职的电脑销售工作。他挣固定的小时工资,但可以通过出售自己出售的电脑的保修金获得奖金。他每周工作20小时。第一周,他卖了8个保修金,挣了220美元。在第二周,他卖了13个保修金,赚了280美元。保罗的小时工资是多少,卖每一份保修单能得到多少额外收入?

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。

7.4 增加或减法的线性系统

章节大纲

常规

增加或减法的线性系统

Solving Linear Systems by Elimination
::通过消除解决线性系统

Solving Linear Systems Using Addition of Equations
::使用增加等量的溶解线性系统

Solving Linear Systems Using Subtraction of Equations
::使用等量减法解决线性系统

Examples
::实例

Review
::回顾

Review (Answers)
::回顾(答复)

章节大纲

常规

增加或减法的线性系统

Solving Linear Systems by Elimination ::通过消除解决线性系统

Solving Linear Systems Using Addition of Equations ::使用增加等量的溶解线性系统

Solving Linear Systems Using Subtraction of Equations ::使用等量减法解决线性系统

Examples ::实例

Review ::回顾

Review (Answers) ::回顾(答复)

Solving Linear Systems by Elimination
::通过消除解决线性系统

Solving Linear Systems Using Addition of Equations
::使用增加等量的溶解线性系统

Solving Linear Systems Using Subtraction of Equations
::使用等量减法解决线性系统

Examples
::实例

Review
::回顾

Review (Answers)
::回顾(答复)