课程： 3.7 以乘数一等数取消变量的溶解系统

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选择节以乘数一等数取消变量的解析系统

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以乘数一等数取消变量的解析系统
Mattie wants to plant some flowers in her yard. She has space for 15 plants. She buys pansies and daisies at her local garden center. The pansies are each $2.75 and the daisies are each $2.00. How many of each does she buy if she spends a total of $35.25?
::Mattie想在院子里种花。她有15种植物的空间。她在当地的园艺中心购买葡萄园和小菊。葡萄园各为2.75美元,小菊每为2.00美元。如果她总共花35.25美元,她要买多少?

Solving Systems by Multiplication
::乘法解决系统

It is not always the case that the coefficients of one variable in a system of linear equations are the same or opposites. When this is not the case, we may be able to multiply one of the equations by a constant so that we have opposite coefficients of one variable and can proceed to solve the system as we have previously done.
::在线性方程式系统中,一个变量的系数并不总是相同或相反。如果情况并非如此,我们也许能够将一个方程式乘以一个常数,这样我们就能有一个变量的相反系数,并像我们以前所做的那样解决系统问题。

Let's solve the following systems using linear combinations.
::让我们用线性组合解决以下系统。

$\begin{aligned} 4 x + y & = 0 \\ x - 3 y & = 26 \end{aligned}$
$\begin{align*}4x+y &= 0\\ x- 3y &= 26\end{align*}$
::4x+y=0x-3y=26

Here we have coefficients of $\begin{align*}y\end{align*}$ that are opposite signs (one is positive and one is negative). We can get opposite values if we multiply the first equation by 3. Be careful; make sure you multiply the entire equation, including the constant, by 3:
::这里的 y 系数是相反的符号( 一个是正数, 一个是负数 ) 。如果我们将第一个方程乘以3, 我们可以得到相反的值。注意小心; 确保您将整个方程, 包括常数乘以3 :

$\begin{aligned} 3 (4 x + y & = 0) \\ 12 x + 3 y & = 0 \end{aligned}$
$\begin{align*}3(4x+y &= 0)\\ 12x+3y &= 0\end{align*}$
::3(4x+y=0)12x+3y=0

Now we can use this new equation in our system to eliminate $\begin{align*}y\end{align*}$ and solve for $\begin{align*}x\end{align*}$ :
::现在我们可以使用我们系统中的这个新方程式来消除y 并解决 x:

$\begin{aligned} 12 x + 3 y = 0 \\ + \underline{x - 3 y = 26} \\ 13 x = 26 \\ x = 2 \end{aligned}$
$\begin{align*}& \quad \ 12x+\cancel{3y}=0\\ & + \ \underline{\;\;\; x-\cancel{3y}=26}\\ & \qquad \quad \ 13x = 26\\ & \qquad \quad \quad \ x=2\end{align*}$
::12x+3y=0+ x-3y=26_ 13x=26 x=2

Now, find $\begin{align*}y\end{align*}$ :
::现在,找到y:

$\begin{aligned} 4 (2) + y & = 0 \\ 8 + y & = 0 \\ y & = - 8 \end{aligned}$
$\begin{align*}4(2)+y &= 0\\ 8+y &= 0\\ y &= -8\end{align*}$
::4(2)+y=08+y=0y8

Solution: (2, -8)
::解决办法2,-8)

Check your answer:
::检查您的答案 :

$\begin{aligned} 4 (2) + (- 8) & = 8 - 8 = 0 \\ (2) - 3 (- 8) & = 2 + 24 = 26 \end{aligned}$
$\begin{align*}4(2)+(-8) &= 8-8=0\\ (2)-3(-8) &= 2+24=26 \end{align*}$

$\begin{align*}^*\end{align*}$ Note that we could have used the other equation to find $\begin{align*}y\end{align*}$ in the final step.
::* 注意我们本可利用另一方程式在最后一步中找到y。

$\begin{align*}^{**}\end{align*}$ From the beginning, we could have multiplied the second equation by -4 instead to cancel out the $\begin{align*}x\end{align*}$ variable.
::从一开始,我们可以将第二个方程乘以 - 4, 以取消 x 变量。

Let's solve the following systems.
::让我们解决以下系统。

$\begin{aligned} 2 x + 5 y & = 1 \\ y & = - 3 x + 21 \end{aligned}$
$\begin{align*}2x+5y &=1\\ y &= -3x+21\end{align*}$
::2x+5y=1y3x+21

For this system, we must first rewrite the second equation in standard form so that we can see how the coefficients compare. If we add $\begin{align*}3x\end{align*}$ to both sides we get:
::对于这个系统,我们必须首先以标准格式重写第二个方程,这样我们可以看到系数是如何比较的。如果我们在两边加3x,我们就会得到:

$\begin{aligned} 2 x + 5 y & = 1 \\ 3 x + y & = 21 \end{aligned}$
$\begin{align*}2x+5y &=1\\ 3x+y &=21\end{align*}$
::2x+5y=13x+y=21

Now, we can see that if we multiply the second equation by -5, the coefficients of $\begin{align*}y\end{align*}$ will be opposites.
::现在,我们可以看到,如果我们把第二个方程乘以5, y的系数就会相反。

$\begin{aligned} 2 x + 5 y = 1 \\ + \underline{- 15 x - 5 y = - 105} \\ - 13 x = - 104 \\ x = 8 \end{aligned}$
$\begin{align*}& \qquad \qquad 2x+\cancel{5y}=1\\ & \quad + \ \ \underline{-15x-\cancel{5y}=-105}\\ & \qquad \qquad \ -13x = -104\\ & \qquad \qquad \qquad \ x = 8\end{align*}$
::2x+5y=1+-15x-5y_105_-13x_104x=8

Now, find $\begin{align*}y\end{align*}$ :
::现在,找到y:

$\begin{aligned} y & = - 3 (8) + 21 \\ y & = - 24 + 21 \\ y & = - 3 \end{aligned}$
$\begin{align*}y &= -3(8)+21\\ y &= -24+21\\ y &= -3\end{align*}$
::y3(8)+21y24+21y3

Solution: (8, -3)
::解决办法: (8, 3)

Check your answer:
::检查您的答案 :

$\begin{aligned} 2 (8) + 5 (- 3) & = 16 - 15 = 1 \\ - 3 = - 3 (8) + 21 & = - 24 + 21 = - 3 \end{aligned}$
$\begin{align*}2(8)+5(-3) &= 16-15 = 1\\ -3=-3(8)+21 &= -24+21=-3 \end{align*}$

Let's solve the following systems.
::让我们解决以下系统。

$\begin{aligned} 4 x - 6 y & = - 12 \\ y & = \frac{2}{3} x + 2 \end{aligned}$
$\begin{align*}4x-6y &= -12\\ y &= \frac{2}{3}x+2\end{align*}$
::4-6y12y=23x+2

Again, we need to rearrange the second equation in this system to get it in standard form. We can do this by subtracting $\begin{align*}\frac{2}{3}x\end{align*}$ on both sides to get the following system:
::我们需要重新排列这个系统中的第二个方程, 才能以标准格式得到它。我们可以通过在两边减去 23x 来做到这一点, 以获得以下系统 :

$\begin{aligned} 4 x - 6 y & = - 12 \\ - \frac{2}{3} x + y & = 2 \end{aligned}$
$\begin{align*}4x-6y &= -12\\ - \frac{2}{3}x+y &= 2\end{align*}$
::4 - 6y 12 - 23x+y=2

Multiply the second equation by 6 to eliminate $\begin{align*}y\end{align*}$ :
::将第二个方程乘以 6 来消除y:

$\begin{aligned} 6 (- \frac{2}{3} x + y = 2) \\ - 4 x + 6 y = 12 \end{aligned}$
$\begin{align*}& \quad 6\left(- \frac{2}{3}x+y =2\right)\\ & \quad \ \ -4x+6y = 12\end{align*}$
::6(- 23x+y=2) - 4x+6y=12

And add it to the first equation.
::加上第一个方程

$\begin{aligned} 4 x - 6 y = - 12 \\ + \underline{- 4 x + 6 y = 12} \\ 0 x = 0 \\ 0 = 0 \end{aligned}$
$\begin{align*}& \qquad \quad 4x-\cancel{6y}= -12\\ & \ \ + \ \underline{-4x+\cancel{6y}=12\;\;\;}\\ & \qquad \qquad \quad 0x=0\\ & \qquad \qquad \quad \ 0=0\end{align*}$
::4 - 6y 12 + - 4x+6y=12_ 0x=0=0

Here, both variables were eliminated and we wound up with 0 = 0. Recall that this is a true statement and thus this system has infinite solutions.
::这里,两个变量都被删除了, 我们最后以 0 = 0 结束。提醒大家注意这是一个真实的语句, 因此这个系统有无限的解决方案。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find how many pansies and daisies Mattie bought at the garden center if the pansies are $2.75 each and the daisies are $2.00 each and she spent a total of $35.25. She has room for 15 plants in her garden.
::早些时候,有人要求你找到在园艺中心买的多少小卖家和小卖家,如果每只小卖家为2.75美元,每只小卖家为2.00美元,她总共花了35.25美元。她在花园里有15种植物。

The system of linear equations represented by this situation is:
::以这种情况为代表的线性方程式系统是:

$\begin{aligned} p + d & = 15 \\ 2.75 p + 2 d & = 35.25 \end{aligned}$
$\begin{align*}p + d &= 15\\ 2.75p + 2d &= 35.25\end{align*}$
::p+d=152.75p+2d=35.25

If we multiply the first of these two equations by $\begin{align*}-2\end{align*}$ , we get a new system of linear equations:
::如果我们将这两个方程式中的第一个方程式乘以-2, 我们就会得到一个新的线性方程式系统:

$\begin{aligned} - 2 p - 2 d & = - 30 \\ 2.75 p + 2 d & = 35.25 \end{aligned}$
$\begin{align*}-2p - 2d &= -30\\ 2.75p + 2d &= 35.25\end{align*}$
::-2p-2d302.75p+2d=35.25

Now we can add these two equations to cancel out the d variable. When we do so, we get:
::现在我们可以添加这两个方程式来取消 d 变量。当我们这样做时, 我们得到:

$\begin{align*}0.75p = 5.25\end{align*}$ or $\begin{align*}p = 7\end{align*}$
::0.75p=5.25或p=7

Finally, we can substitute $\begin{align*}p = 7\end{align*}$ into either of our original equations to get the value of d .
::最后,我们可以用p=7 来替代我们原来的方程中的任何一个方程来获得d的值。

$\begin{align*}7 + d = 15\end{align*}$ or $\begin{align*}d = 8\end{align*}$
::7+d=15或d=8

Therefore Mattie buys 7 pansies and 8 daisies.
::所以Mattie买了7个罐子和8个菊花

Example 2
::例2

Solve the following systems using linear combinations.
::使用线性组合解决以下系统。

$\begin{aligned} 3 x + 12 y & = - 3 \\ - x - 5 y & = 0 \end{aligned}$
$\begin{align*}3x+12y &= -3\\ -x-5y &= 0\end{align*}$
::3x+12y3-x-5y=0

In this problem we can just multiply the second equation by 3 to get coefficients of $\begin{align*}x\end{align*}$ which are opposites: $\begin{align*}3(-x-5y=0)\Rightarrow -3x-15y =0\end{align*}$
::在此问题上,我们只需将第二个方程乘以3即可获得相反的x系数:3(3-x-5y=0)3x-15y=0。

$\begin{aligned} 3 x + 12 y = - 3 \\ \underline{- 3 x - 15 y = 0} \\ - 3 y = - 3 \\ y = 1 \end{aligned}$
$\begin{align*}&\ \ \cancel{3x}+12y=-3\\ &\underline{-\cancel{3x}-15y=0\;\;\;}\\ & \qquad \ -3y=-3\\ & \qquad \qquad y=1\end{align*}$
::3x+12y3-3x-15y=0_-3y3y=1

Now we can find $\begin{align*}x\end{align*}$ :
::现在我们可以找到x:

$\begin{aligned} 3 x + 12 (1) & = - 3 \\ 3 x + 12 & = - 3 \\ 3 x & = - 15 \\ x & = - 5 \end{aligned}$
$\begin{align*}3x+12(1) &= -3\\ 3x+12 &= -3\\ 3x &= -15\\ x &= -5\end{align*}$
::3x+12(1)33x+1233x15x5

Solution: (-5, 1)
::解决办法5,1)

Example 3
::例3

Solve the following systems using linear combinations.
::使用线性组合解决以下系统。

$\begin{aligned} 0.75 x + 5 y & = 0 \\ 0.25 x - 9 y & = 0 \end{aligned}$
$\begin{align*}0.75x+5y &= 0\\ 0.25x-9y &= 0\end{align*}$
::0.75x+5y=00.25x-9y=0

For this system, we need to multiply the second equation by -3 to get coefficients of $\begin{align*}x\end{align*}$ which are opposites: $\begin{align*}-3(0.25x-9y=0) \Rightarrow -0.75x+27y=0\end{align*}$
::对于这个系统,我们需要将第二个方程乘以 - 3, 才能取得相反的 x 系数 : - 3( 0.25x- 9y=0) = 0.75x+27y=0

$\begin{aligned} 0.75 x + 5 y = 0 \\ - \underline{0.75 x - 9 y = 0} \\ - 4 y = 0 \\ y = 0 \end{aligned}$
$\begin{align*}& \quad \cancel{0.75x}+5y=0\\ & - \underline{\cancel{0.75x}-9y=0}\\ & \qquad \quad \ -4y=0\\ & \qquad \qquad \quad y=0\end{align*}$
::0.75x+5y=0-0.75x-9y=0_-4y=0y=0

Now we can find $\begin{align*}x\end{align*}$ :
::现在我们可以找到x:

$\begin{aligned} 0.75 x + 5 (0) & = 0 \\ 0.75 x & = 0 \\ x & = 0 \end{aligned}$
$\begin{align*}0.75x+5(0) &= 0\\ 0.75x &= 0\\ x &= 0\end{align*}$
::0.75x+5(0)=00.75x=0x=0x=0

Solution: (0, 0)
::解决方案: (0, 0)

Example 4
::例4

Solve the following systems using linear combinations.
::使用线性组合解决以下系统。

$\begin{aligned} x - 3 y & = 5 \\ y & = \frac{1}{3} x + 8 \end{aligned}$
$\begin{align*}x-3y &= 5\\ y &= \frac{1}{3}x+8\end{align*}$
::x-3y=5y=13x+8

This time we need to rewrite the second equation in standard form:
::这次我们需要以标准格式重写第二个方程式:

$\begin{aligned} x - 3 y & = 5 \\ - \frac{1}{3} x + y & = 8. \end{aligned}$
$\begin{align*}x-3y &= 5\\ - \frac{1}{3}x+y &= 8.\end{align*}$
::x-3y=5-13x+y=8。

Now we can multiply the second equation by 3 to get coefficients of $\begin{align*}x\end{align*}$ that are opposites:
::现在,我们可以将第二个方程乘以 3 来获得相反的 x 系数 :

$\begin{align*}3 \left(-\frac{1}{3}x+y=8\right) \Rightarrow -x+3y=24\end{align*}$ , Now our system is:
:3 - 13x+y=8)\\ x+3y=24)\\\ x+3y=24\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\我们的系统是:

$\begin{aligned} x - 3 y & = 5 \\ - x + 3 y & = 24 \end{aligned}$
$\begin{align*}x-3y &= 5\\ -x+3y &= 24\end{align*}$
::x-3y=5-x+3y=24

When we add these equations together, both variables are eliminated and the result is 0 = 29, which is an untrue statement. Therefore, this system has no solution.
::当我们把这些方程式加在一起时,这两个变量都被删除,结果为0=29,这是不真实的陈述。因此,这个系统没有解决办法。

Review
::回顾

Solve the following systems using linear combinations.
::使用线性组合解决以下系统。

.

$\begin{aligned} x - 7 y & = 27 \\ 2 x + y & = 9 \end{aligned}$
$\begin{align*}x-7y &= 27\\ 2x+y &= 9\end{align*}$

.

$\begin{aligned} x + 3 y & = 31 \\ 3 x - 5 y & = - 5 \end{aligned}$
$\begin{align*}x+3y &= 31\\ 3x-5y &= -5\end{align*}$

.

$\begin{aligned} 10 x + y & = - 6 \\ - 7 x - 5 y & = - 13 \end{aligned}$
$\begin{align*}10x+y &= -6\\ -7x-5y &= -13\end{align*}$

.

$\begin{aligned} 2 x + 4 y & = 18 \\ x - 5 y & = 9 \end{aligned}$
$\begin{align*}2x+4y &= 18\\ x-5y &= 9\end{align*}$

.

$\begin{aligned} 2 x + 6 y & = 8 \\ 3 x + 2 y & = - 23 \end{aligned}$
$\begin{align*}2x+6y &= 8\\ 3x+2y &= -23\end{align*}$

.

$\begin{aligned} 12 x - y & = 2 \\ 2 x + 5 y & = 21 \end{aligned}$
$\begin{align*}12x-y &= 2\\ 2x+5y &= 21\end{align*}$

.

$\begin{aligned} 2 x + 4 y & = 24 \\ - 3 x - 2 y & = - 26 \end{aligned}$
$\begin{align*}2x+4y &= 24\\ -3x-2y &= -26\end{align*}$

.

$\begin{aligned} 3 x + 2 y & = 19 \\ 5 x + 4 y & = 23 \end{aligned}$
$\begin{align*}3x+2y &= 19\\ 5x+4y &= 23\end{align*}$

.

$\begin{aligned} 3 x - 9 y & = 13 \\ x - 3 y & = 7 \end{aligned}$
$\begin{align*}3x-9y &= 13\\ x-3y &= 7\end{align*}$

.

$\begin{aligned} 8 x + 2 y & = - 4 \\ 3 y & = - 16 x + 2 \end{aligned}$
$\begin{align*}8x+2y &= -4\\ 3y &= -16x+2\end{align*}$

.

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

.

$\begin{aligned} 10 x + 6 y & = - 24 \\ y & = - \frac{5}{3} x - 4 \end{aligned}$
$\begin{align*}10x+6y &= -24\\ y &= - \frac{5}{3}x-4\end{align*}$

.

$\begin{aligned} \frac{1}{3} x - \frac{2}{3} y & = - 8 \\ \frac{1}{2} x - \frac{1}{3} y & = 12 \end{aligned}$
$\begin{align*}\frac{1}{3}x-\frac{2}{3}y &= -8\\ \frac{1}{2}x-\frac{1}{3}y &= 12\end{align*}$

.

$\begin{aligned} 6 x - 10 y & = - 8 \\ y & = - \frac{3}{5} x \end{aligned}$
$\begin{align*}6x-10y &= -8\\ y &= - \frac{3}{5}x\end{align*}$

.

$\begin{aligned} 4 x - 14 y & = - 52 \\ y & = \frac{2}{7} x + 3 \end{aligned}$
$\begin{align*}4x-14y &= -52\\ y &= \frac{2}{7}x+3\end{align*}$

Set up and solve a system of linear equations for each of the following word problems.
::为以下每个字问题建立和解决线性方程式系统。

Lia is making a mixture of Chlorine and water in her science class. She needs to make 13 ml of a 60% chlorine solution from a solution that is 35% chlorine and a second solution which is 75% chlorine. How many milliliters of each solution does she need?
::Lia在理科课上混合了氯和水。她需要从35%氯的溶液和75%氯的第二个溶液中制成13毫升的60%氯溶液。她需要多少毫升的溶液?

Chelsea and Roberto each sell baked goods for their club’s fundraiser. Chelsea sells 13 cookies and 7 brownies and collects a total of $11.75. Roberto sells 10 cookies and 14 brownies and collects a total of $15.50. How much did they charge for the cookies and the brownies?
::切尔西和罗伯托各自为俱乐部的筹款者出售面包食品。切尔西出售13个饼干和7个巧克力蛋糕,共收11.75美元。罗伯托出售10个饼干和14个巧克力蛋糕,共收15.50美元。饼干和蛋糕的收费是多少?

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。

章节大纲

常规

以乘数一等数取消变量的解析系统

Solving Systems by Multiplication ::乘法解决系统

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Solving Systems by Multiplication
::乘法解决系统

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)