节: 三等和三未知系统 | 8.2 三种等式和三种未知的系统

章节大纲

Later, you will learn about matrices and how to row reduce which will allow you to solve systems of equations in a new way. In order to set you up so that using matrices is logical and helpful, it is important to first solve a few systems of three equations using a very specific type of variable elimination.
::稍后,您将了解矩阵和如何排行缩减,这样您就可以用新的方式解析方程式系统。为了设置您,以便使用矩阵是合乎逻辑的和有用的,首先必须使用非常具体的变数去除类型解决三个方程式的几个系统。

When solving systems, what are you allowed to do to each equation ?
::当解决系统时, 你允许对每个方程式做什么?

Solving Systems of Equations with Three Unknowns
::三个未知方的溶解系统

A system of three equations with three unknowns represents three planes in three dimensional space. When solving the system, you are figuring out how the planes intersect. One way that three planes could intersect is in a point:
::由三个方程式组成的系统有三个未知方程式, 代表三维空间中的三平面。当解开这个系统时, 您正在了解三平面是如何交叉的。一种方式是三平面可以交叉到一个点上 :

A system of equations that has at least one solution is called a consistent system .
::至少有一种解决办法的方程式系统称为连贯一致的系统。

It is also possible for two or more planes to be parallel or each pair of planes to intersect in a line. In either of these cases the three planes do not intersect at a single point and the system is said to have no solution. A system of equations with no solutions is called an inconsistent system . If the three planes intersect at a line or a plane, there are an infinite number of solutions.
::两个或两个以上的平面也可以平行或每对平面在一条线上交叉。在这两种情况下,任何一种情况下,三架飞机在一个点上不交叉,而系统据说没有解决办法。一个没有解决办法的方程式系统被称为一个不一致的系统。如果三架飞机在一条线上或一平面上交叉,则有无限数量的解决方法。

The following system of equations has the solution (1, 3, 7). You can verify this by substituting 1 for $\begin{aligned} x \end{aligned}$ , 3 for $\begin{aligned} y \end{aligned}$ , and 7 for $\begin{aligned} z \end{aligned}$ into each equation.
::下列方程式系统有解决办法(1,3,7),每个方程式以1取代1,3取代x,3取代y,7取代z。

\begin{aligned} x + 2 y - z & = 0 \\ 7 x - 0 y + z & = 14 \\ 0 x + y + z & = 10 \end{aligned}

::x+2y-z=07x-0y+z=140x+y+z=10

One thing to be mindful of when given a system of equations is whether or not the equations are linearly independent . Three equations are linearly independent if each equation cannot be produced by a linear combination of the other two. Remember that a linear combination means that one equation can be written as the sum of multiples of the others.
::当给定一个方程式系统时,需要注意的一件事是,这些方程式是否线性独立。如果每个方程式不能用其他两个方程式的线性组合产生,三个方程式就线性独立。记住线性组合意味着一个方程式可以写成为其他方程式的倍数之和。

When solving a system of three equations and three variables, there are a few general guidelines that can be helpful:
::在解决由三个方程式和三个变量组成的系统时,有一些一般准则可以起到帮助作用:

Start by trying to eliminate the first variable in the second row.
::从尝试删除第二行的第一个变量开始。

Next eliminate the first and second variables in the third row. This will create zero coefficients in the lower right hand corner.
::下一步删除第三行的第一和第二变量。这将在右下角的右下角产生零系数。

Repeat this process for the upper right hand corner and you should end up with a very nice diagonal indicating what $\begin{aligned} x, y \end{aligned}$ and $\begin{aligned} z \end{aligned}$ equal.
::重覆右上角的此进程, 最终您应该有一个非常好的对角, 显示 x, y 和 z 等值。

Take the system of equations mentioned above:
::采用上述方程式系统:

\begin{aligned} x + 2 y - z & = 0 \\ 7 x - 0 y + z & = 14 \\ 0 x + y + z & = 10 \end{aligned}

::x+2y-z=07x-0y+z=140x+y+z=10

There are a number of ways to solve this system. Common techniques involve swapping rows, dividing and multiplying a row by a constant and adding or subtracting a multiple of one row to another.
::解决这个系统有多种方法。常见技术包括换行、以常数分隔和乘以一行、加上或减去一行的倍数。

Step 1: Swap rows 2 and 3. Change -0 to +0.
::第1步:交换第2行和第3行,更改-0至+0。

$\begin{aligned} x + 2 y - z & = 0 \\ 0 x + y + z & = 10 \\ 7 x + 0 y + z & = 14 \end{aligned}$

::x+2y-z=00x+y+z=107x+0y+z=14

Step 2: Subtract 7 times row 1 from row 3, then replace row 3.
::第2步:将第3行第1行减7倍,然后取代第3行。

$\begin{aligned} x + 2 y - z & = 0 \\ 0 x + y + z & = 10 \\ 0 x - 14 y + 8 z & = 14 \end{aligned}$

::x+2y-z=00x+y+z=100x-14y+8z=14

Step 3: Add 14 times row 2 to row 3 and then replace row 3.
::第3步:将第2行第14次加到第3行,然后取代第3行第14次。

$\begin{aligned} x + 2 y - z & = 0 \\ 0 x + y + z & = 10 \\ 0 x + 0 y + 22 z & = 154 \end{aligned}$

::x+2y-z=00x+y+z=100x+0y+22z=154

Step 4: Divide row 3 by 22.
::第4步:将第3行除以22。

$\begin{aligned} x + 2 y - z & = 0 \\ 0 x + y + z & = 10 \\ 0 x + 0 y + z & = 7 \end{aligned}$

::x+2y-z=00x+y+z=100x+0y+z=7

Step 5: Subtract row 3 from row 2 and then replace row 2.
::第5步:将第3行从第2行减为第3行,然后取代第2行。

$\begin{aligned} x + 2 y - z & = 0 \\ 0 x + y + 0 z & = 3 \\ 0 x + 0 y + z & = 7 \end{aligned}$

::x+2y-z=00x+y+0z=30x+0y+z=7

Step 6: Add row 3 to row 1 and then replace row 1.
::第6步:将第3行添加到第1行,然后替换第1行。

$\begin{aligned} x + 2 y + 0 z & = 7 \\ 0 x + y + 0 z & = 3 \\ 0 x + 0 y + z & = 7 \end{aligned}$

::x+2y+0z=70x+y+0z=30x+0y+z=7

Step 7: Subtract 2 times row 2 from row 1 and then replace row 1.
::第7步:从第1行中减2倍于第2行,然后取代第1行。

$\begin{aligned} x + 0 y + 0 z & = 1 \\ 0 x + y + 0 z & = 3 \\ 0 x + 0 y + z & = 7 \end{aligned}$

::x+0y+0z=10x+y+0z=30x+0y+z=7

The solution to the system is (1, 3, 7) exactly as stated earlier.
::该系统的解决方案是(1,3,7),正如前所述。

Examples
::实例

Example 1
::例1

Earlier, you were asked what you are allowed to do when solving a system of three equations. When solving a system of three equations with three unknowns, you are allowed to add and subtract rows, swap rows and scale rows. These three operations should allow you to eliminate the coefficients of the variables in a systematic way.
::早些时候, 有人询问您在解决三个方程式的系统时可以做什么。当解决三个方程式的系统时, 三个方程式有三个未知数, 允许您增减行、互换行和缩放行。这三个操作应该允许您系统地消除变量的系数。

Example 2
::例2

Is the following system linearly independent or dependent ? How do you know?
::下列系统是线性独立的还是依赖性的?你怎么知道的?

\begin{aligned} 3 x + 2 y + z & = 8 \\ x + y + z & = 3 \\ 5 x + 4 y + 3 z & = 14 \\ 6 x + 6 y + 6 z & = 18 \end{aligned}

::3x+2y+z=8x+y+z=35x+4y+3z=146x+6y+6z=18

With four equations and three unknowns there must be at least one dependent equation. The simplest method of seeing linearly dependence is to notice that one equation is just a multiple of the other. In this case the fourth equation is just six times the second equation and so it is dependent.
::四个方程式和三个未知方程式必须至少有一个依附方程式。观察线性依赖的最简单方法是注意一个方程式只是另一个方程式的倍数。在这种情况下,第四个方程式是第二个方程式的六倍,因此它取决于第二个方程式。

Most people will not notice that the third equation is also dependent. It is common to start doing a problem and notice somewhere along the way that all the variables in a row disappear. This means that the original equations were dependent. In this case, the third equation is the first equation plus two times the second equation. This means they are dependent.
::大多数人不会注意到第三个方程是否也取决于第三个方程。通常会开始做一个问题, 并发现在某处一行中的所有变量都消失。这意味着原始方程是依赖的。在这种情况下, 第三个方程是第一个方程加上第二个方程的两倍。这意味着它们是依赖的。

Example 3
::例3

Reduce the following system to a system of two equations and two unknowns.
::将以下系统减为由两个方程式和两个未知方程式组成的系统。

\begin{aligned} 3 x + 2 y + z & = 7 \\ 4 x + 0 y + z & = 6 \\ 6 x - y + 0 z & = 5 \end{aligned}

::3x+2y+z=74x+0y+z=66x-y+0z=5

Strategically swapping rows so that the zero coefficients do not live on the diagonal is a clever starting move.
::战略互换行让零系数不生活在对角线上,这是一个明智的起始动作。

Step 1: Swap rows 2 and 3.
::第1步:交换第2行和第3行。

$\begin{aligned} 3 x + 2 y + z & = 7 \\ 6 x - y + 0 z & = 5 \\ 4 x + 0 y + z & = 6 \end{aligned}$

::3x+2y+z=76x-y+0z=54x+0y+z=6

Step 2: Scale row 3 by a factor of 3. Subtract 2 times row 1 from row 2 and replace row 2.
::第2步:将第3行缩放3乘以3,将第2行第1行减为第2行第2倍,替换第2行第2行。

$\begin{aligned} 3 x + 2 y + z & = 7 \\ 0 x - 5 y - 2 z & = - 9 \\ 12 x + 0 y + 3 z & = 18 \end{aligned}$

::3x+2y+z=70x-5y-2z @%912x+0y+3z=18

Step 3: Subtract 4 times row 1 from row 3 and replace row 3.
::第3步:将第3行第1行减4倍,替换第3行。

$\begin{aligned} 3 x + 2 y + z & = 7 \\ 0 x - 5 y - 2 z & = - 9 \\ 0 x - 8 y - z & = - 10 \end{aligned}$

::3x+2y+z=70x-5y-2z90x-8y-z10

Step 4: Scale the second row by 8 and the third row by 5.
::第4步:第二行缩小8,第三行缩小5。

$\begin{aligned} 3 x + 2 y + z & = 7 \\ 0 x - 40 y - 16 z & = - 72 \\ 0 x - 40 y - 5 z & = - 50 \end{aligned}$

::3x+2y+z=70x-40y-16z @%720x-40y-5z50

Step 5: Subtract row 2 from row 3 and replace row 3.
::第5步:从第3行中减去第2行,替换第3行。

$\begin{aligned} 3 x + 2 y + z & = 7 \\ 0 x - 40 y - 16 z & = - 72 \\ 0 x + 0 y + 11 z & = + 22 \end{aligned}$

::3x+2y+z=70x-40y-16z+720x+0y+11z+22

Step 6: Divide row 3 by 11.
::步骤6:将第3行除以11。

$\begin{aligned} 3 x + 2 y + z & = 7 \\ 0 x - 40 y - 16 z & = - 72 \\ 0 x + 0 y + z & = 2 \end{aligned}$

::3x+2y+z=70x-40y-16z+720x+0y+z=2

Now that $\begin{aligned} z = 2 \end{aligned}$ , rewrite the system so it becomes a system of two equations and two unknowns.
::现在,z=2, 重写系统, 使它变成一个由两个方程和两个未知的系统。

\begin{aligned} 3 x + 2 y + 2 & = 7 \\ 0 x - 40 y - 32 & = - 72 \end{aligned}

::3x+2y+2=70x-40y-3272

Solving the second row shows that $\begin{aligned} y = 1 \end{aligned}$ which can be used to determine that $\begin{aligned} x = 1. \end{aligned}$
::解析第二行显示,y=1可用于确定 x=1。

Example 4
::例4

When Kaitlyn went to the store with ten dollars she saw that she had some choices about what to buy. She could get one apple, one onion and one basket of blueberries for 9 dollars. She could get two apples and two onions for 10 dollars. She could also get two onions and one basket of blueberries for 10 dollars. Write and solve a system of equations with variables $\begin{aligned} a, o \end{aligned}$ and $\begin{aligned} b \end{aligned}$ representing each of the three things she can buy.
::Kaitlyn带着10美元去商店的时候,她看到她对买什么有一些选择。她可以买一个苹果、一个洋葱和一个蓝莓篮子,9美元。她可以买两个苹果和两个洋葱,10美元。她也可以买两个洋葱和一个蓝莓篮子,10美元。写和解决一个包含变量a、o和b的方程式系统,代表她可以买三个东西中的每一个。

Here is the system of equations:
::以下是方程式系统:

\begin{aligned} a + o + b & = 9 \\ 2 a + 2 o & = 10 \\ 2 o + b & = 10 \end{aligned}

::a+o+b=92a+2o=102o+b=10

Rewrite the system using $\begin{aligned} x, y \end{aligned}$ and $\begin{aligned} z \end{aligned}$ so that $\begin{aligned} o \end{aligned}$ and 0 do not get mixed up. Include coefficients of 0 so that each column represents one variable.
::使用 x、y 和 z 重写系统, 使 o 和 0 不被混合。包含 0 的系数, 这样每列代表一个变量。

Step 1: Rewrite
::第1步:重写

$\begin{aligned} 1 x + 1 y + 1 z & = 9 \\ 2 x + 2 y + 0 z & = 10 \\ 0 x + 2 y + 1 z & = 10 \end{aligned}$

::1x+1y+1z=92x+2y+0z=100x+2y+1z=10

Step 2: Subtract 2 times row 1 from row 2 and replace row 2.
::第2步:从第2行减2倍于第1行,替换第2行。

$\begin{aligned} 1 x + 1 y + 1 z & = 9 \\ 0 x + 0 y - 2 z & = - 8 \\ 0 x + 2 y + 1 z & = 10 \end{aligned}$

::1x+1y+1z=90x+0y=2z=80x+2y+1z=10

Step 3: Divide row 2 by -2.
::第3步:将第2行除以-2。

$\begin{aligned} 1 x + 1 y + 1 z & = 9 \\ 0 x + 0 y + 1 z & = 4 \\ 0 x + 2 y + z & = 10 \end{aligned}$

::1x+1y+1z=90x+0y+1z=40x+2y+z=10

At this point you can see from the second equation that $\begin{aligned} z = 4 \end{aligned}$ . From the third equation, $\begin{aligned} 2 y + 4 = 10 \end{aligned}$ , so $\begin{aligned} y = 3 \end{aligned}$ . Finally you can see from the first equation that $\begin{aligned} x + 3 + 4 = 9 \end{aligned}$ so $\begin{aligned} x = 2 \end{aligned}$ . Apples cost 2 dollars each, onions cost 3 dollars each and blueberries cost 4 dollars each.
::此时,您可以从第二个方程 z=4 中看到。从第三个方程, 2y+4=10, so y=3, 最后您可以从第一个方程看到 x+3+4=9, so x=2 苹果每个费用2美元,洋葱每个费用3美元,蓝莓每个费用4美元。

Example 5
::例5

Show that the following system is dependent.
::显示以下系统是否依附于以下系统。

\begin{aligned} x + y + z & = 9 \\ x + 2 y + 3 z & = 22 \\ 2 x + 3 y + 4 z & = 31 \end{aligned}

::x+y+z=9x+2y+3z=222x+3y+4z=31

You could notice that the third equation is simply the sum of the other two. What happens when you do not notice and try to solve the system as if it were independent ?
::您可以注意到第三个方程仅仅是另外两个方程的总和。当您不注意并试图解决系统时会怎样,就像它独立了一样?

Step 1: Rewrite the system.

$\begin{aligned} x + y + z & = 9 \\ x + 2 y + 3 z & = 22 \\ 2 x + 3 y + 4 z & = 31 \end{aligned}$

::第1步:重写系统.x+y+z=9x+2y+3z=222x+3y+4z=31

Step 2: Subtract 2 times row 1 from row 3 and replace row 3.
::第2步:从第3行减2倍于第1行,替换第3行。

$\begin{aligned} x + y + z & = 9 \\ x + 2 y + 3 z & = 22 \\ 0 x + 1 y + 2 z & = 13 \end{aligned}$

::x+y+z=9x+2y+3z=220x+1y+2z=13

Step 3: Subtract row 1 from row 2 and replace row 2.
::第3步:将第1行从第2行减为第1行,替换第2行。

$\begin{aligned} x + y + z & = 9 \\ 0 x + 1 y + 2 z & = 13 \\ 0 x + 1 y + 2 z & = 13 \end{aligned}$

::x+y+z=90x+1y+2z=130x+1y+2z=13

At this point when you subtract row 2 from row 3, all the coefficients in row 3 disappear. This means that you will end up with the following system of only two equations and three unknowns. Since the unknowns outnumber the equations, the system does not have a solution of one point.
::当您从第3行中减去第2行时, 第3行中的所有系数都会消失。这意味着您最终会出现以下两个方程式和三个未知方程式的系统。由于未知数超过方程式, 系统无法解决一个点。

\begin{aligned} x + y + z & = 9 \\ 0 x + 1 y + 2 z & = 13 \end{aligned}

::x+y+z=90x+1y+2z=13

Remember this!

A consistent system has at least one solution, while an inconsistent system has no solutions.
::一致的系统至少有一种解决办法,而不一致的系统则没有解决办法。

Three equations are linearly independent if each equation cannot be produced by a linear combination of the other two.
::如果每个方程式不能用其他两个方程式的线性组合产生,则三个方程式是线性独立的。

Common techniques for solving systems of equations involve swapping rows, dividing and multiplying a row by a constant, and adding or subtracting a multiple of one row to another.
::解决方程式系统的共同技术包括互换行数、以常数将一行除以乘以一行、将一行乘以另一行相加或减以一行的倍数。

Review
::回顾

1. An equation with three variables represents a plane in space. Describe all the ways that three planes could interact in space.
::1. 具有三个变量的方程式代表空间中的一架飞机,描述三架飞机在空间中相互作用的所有方式。

2. What does it mean for equations to be linearly dependent?
::2. 公式线性依赖意味着什么?

3. How can you tell that a system is linearly dependent?
::3. 你怎么知道一个系统线性依赖?

4. If you have linearly independent equations with four unknowns, how many of these equations would you need in order to get one solution?
::4. 如果你有四种未知的线性独立方程式,那么要找到一个解决方案,你需要多少这些方程式?

5. Solve the following system of equations:
::5. 解决下列方程式系统:

\begin{aligned} 3 x - 4 y + z & = - 17 \\ 6 x + y - 3 z & = 4 \\ - x - y + 5 z & = 16 \end{aligned}

::3x-4y+z176x+y-3z=4-x-y+5z=16

6. Show that the following system is dependent:
::6. 显示以下系统是否依赖:

\begin{aligned} 2 x - 2 y + z & = 5 \\ 6 x + y - 3 z & = 2 \\ 4 x + 3 y - 4 z & = - 3 \end{aligned}

::2 - 2y+z=56x+y-3z=24x+3y-4z3

7. Solve the following system of equations:
::7. 解决下列方程式系统:

\begin{aligned} 4 x + y + z & = 15 \\ - 2 x + 3 y + 4 z & = 38 \\ - x - y + 3 z & = 16 \end{aligned}

::4x+y+z=15-2x+3y+4z=38-x-y+3z=16

8. Solve the following system of equations:
::8. 解决下列方程式系统:

\begin{aligned} 3 x - 2 y + 3 z & = 6 \\ x + 3 y - 3 z & = - 14 \\ - x + y + 5 z & = 22 \end{aligned}

::3x-2y+3z=6x+3y-3z_3z_14-x+y+5z=22

9. Solve the following system of equations:
::9. 解决下列方程式系统:

\begin{aligned} 3 x - y + z & = - 10 \\ 6 x - 2 y + 2 z & = - 20 \\ - x - y + 4 z & = 12 \end{aligned}

::3 - y+zZZ106x-2y+2z20 - x-y+4z=12

10. Solve the following system of equations:
::10. 解决下列方程式系统:

\begin{aligned} x - 3 y + 6 z & = - 30 \\ 4 x + 2 y - 3 z & = 18 \\ - 8 x - 3 y + 2 z & = - 22 \end{aligned}

::x-3y+6z304x+2y-3z=18-8x-3y+2z22

11. Solve the following system of equations.
::11. 解决下列方程式系统。

\begin{aligned} x + 2 y + 2 z + w & = 5 \\ 2 x + y + 2 z - 0 w & = 5 \\ 3 x + 3 y + 3 z + 2 w & = 12 \\ x + 0 y + z + w & = 1 \end{aligned}

::x+2y+2z+w=52x+y+2z=0w=53x+3y+3z+2w=12x+0y+z+w=1

A parabola goes through (3, -9.5), (6, -32), and (-4, 8).
::抛物线通过(3,9.5)、(6,32)和(4,8)。

12. Write a system of equations that you could use to solve to find the equation of the parabola. Hint: Use the general equation $\begin{aligned} A x^{2} + B x + C = y \end{aligned}$ .
::12. 写入一个可以用来解析的方程式系统,以找到抛物线的方程式。提示:使用一般方程式 Ax2+Bx+C=y。

13. Solve the system of equations from #12.
::13. 从第12号解决方程式系统。

A parabola goes through (-2, 3), (2, 19), and (1, 6).
::抛物线通过(-2,3,2,19)和(1,6)。

14. Write a system of equations that you could use to solve to find the equation of the parabola. Hint: Use the general equation $\begin{aligned} A x^{2} + B x + C = y \end{aligned}$ .
::14. 写出一个可以用来解析的方程式系统,以找到抛物线的方程式。提示:使用一般方程式 Ax2+Bx+C=y。

15. Solve the system of equations from #14.
::15. 从14号解决方程式系统。

Review (Answers)
::回顾(答复)

To see the answer key for this book, go to the and click on the Answer Key under the ' ' option.
::要查看本书的答案键, 请在“ ” 选项下点击答案键。

章节大纲

Solving Systems of Equations with Three Unknowns ::三个未知方的溶解系统

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)