3.4 使用替代物的一次性解决办法的溶解系统

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  使用替代物的单一解决办法的解决系统

  Rex and Carl are making a mixture in science class. They need to have 12 ounces of a 60% saline solution. To make this solution they have a 20% saline solution and an 80% saline solution. How many ounces of each do they need to make the correct mixture?

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  Systems with One Solution Using Substitution
  ::使用替代的一解决方案系统

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  In the substitution method we will be looking at the two equations and deciding which variable is easiest to solve for so that we can write one of the equations as $\begin{align*}x=\end{align*}$ or $\begin{align*}y=\end{align*}$ . Next we will replace either the $\begin{align*}x\end{align*}$ or the $\begin{align*}y\end{align*}$ accordingly in the other equation . The result will be a solve-able equation with only one variable.
  ::在替代方法中,我们将查看这两个方程式,并决定最容易解决的变量,以便我们可以将其中一个方程式写成 x= 或 Y= 。接下来我们将在另一个方程式中相应替换 x 或 y。结果将是一个只有一个变量的可解决方程式。

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  Let's solve the systems below using substitution.
  ::让我们用替代来解决下面的系统

  1.     

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  $$\begin{align*}2x+y &= 12\\ \text{-}3x-5y &= \text{-}11\end{align*}$$
  ::2x+y=12-3x-5y=11

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  The first step is to look for a variable that is easy to isolate. In other words, does one of the variables have a coefficient of 1? Yes, that variable is the $\begin{align*}y\end{align*}$ in the first equation. So, start by solving for (isolating)  $\begin{align*}y: \ \ y=\text{-}2x+12.\end{align*}$
  ::第一步是寻找一个容易分离的变量。 换句话说, 变量之一的系数是否为 1 ? 是的, 该变量是第一个方程中的 y。 所以, 从解决( 孤立) y: y= 2x+12 开始 。

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  This expression can then be used to replace the $\begin{align*}y\end{align*}$ in the other equation to  solve for $\begin{align*}x:\end{align*}$
  ::此表达式可用于替换其他方程式中的 Y, 以解析 x:

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  $$\begin{align*}\text{-}3x-5(\text{-}2x+12) &= \text{-}11\\ \text{-}3x+10x-60 &= \text{-}11\\ 7x-60 &= \text{-}11\\ 7x &= 49\\ x &= 7\end{align*}$$
  ::-3x-5(-2x+12)=-11-3x+10x-60=-117x-60=-117x=-117x=-117x=-49x=7

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  Now that we have found $\begin{align*}x\end{align*}$ , we can use this value in our expression to find $\begin{align*}y:\end{align*}$
  ::既然我们找到了 x, 我们可以在表达式中使用这个值来查找 y:

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  $$\begin{align*} y &= \text{-}2(7)+12\\ y &= \text{-}14+12\\ y &= \text{-}2\end{align*}$$
  ::y=-2(7)+12y=-14+12y=-2

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  Recall that the solution to a linear system is a point in the coordinate plane where the two lines intersect. Therefore , our answer should be written as a point: (7, -2). You can check your answer by substituting this point into both equations to make sure that it satisfies them:
  ::回顾线性系统的解决方案是坐标平面上两条线交叉的一个点。 因此, 我们的答案应该写成一个点 : (7, - 2) 。 您可以将这个点替换成两个方程式来检查答案, 以确保它满足 :

  $$\begin{align*}2(7)+ \text{-}2 &= 14-2=12\\ \text{-}3(7)-5(\text{-}2) &= \text{-}21+10 = \text{-}11 \end{align*}$$

  2.    

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  $$\begin{align*}2x+3y &= 13\\ x+5y &= \text{-}4\end{align*}$$
  ::2x+3y=13x+5y=-4

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  In the last problem , $\begin{align*}y\end{align*}$ was the easiest variable to isolate. Is that the case here? No, this time, $\begin{align*}x\end{align*}$ is the variable with a coefficient of 1. It is easy to fall into the habit of always isolating $\begin{align*}y\end{align*}$ since you have done it so much to write equation in slope-intercept form . Try to avoid this and look at each system to see which variable is easiest to isolate. Doing so will help reduce your work.
  ::在最后一个问题中, y 是最容易分离的变量 。 情况是这样吗 ? 不, 这次, x 是系数为 1 的变量 。 由于您已经做了很多工作, 用斜坡界面写了方程式。 尝试避免这一点, 并查看每个系统 看看哪个变量最容易分离 。 这样做有助于减少您的工作 。

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  Solving the second equation for $\begin{align*}x\end{align*}$ gives: $\begin{align*}x = \text{-}5y-4.\end{align*}$
  ::解析 x 的第二个方程式时给出: x=-5y-4 。

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  This expression can be used to replace the $\begin{align*}x\end{align*}$ in the other equation and solve for $\begin{align*}y:\end{align*}$
  ::此表达式可用于替换其他方程式中的 x, 并解决 y:

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  $$\begin{align*}2(\text{-}5y-4)+3y &= 13\\ \text{-}10y-8+3y &= 13\\ \text{-}7y-8 &= 13\\ \text{-}7y &= 21\\ y &= \text{-}3\end{align*}$$
  ::2(5-5y-4)+3y=13-10y-8+3y=13-7y-8=13-7y=13-7y=21y=3)

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  Now that we have found $\begin{align*}y\end{align*}$ , we can use this value in our expression to find $\begin{align*}x\end{align*}$ :
  ::既然我们找到了y, 我们可以在表达式中使用这个值来找到x:

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  $$\begin{align*}x &= \text{-}5(\text{-}3)-4\\ x &= 15-4\\ x &= 11\end{align*}$$
  ::x=5(-3)-4x=15-4x=11

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  So, the solution to this system is (11, -3). Don’t forget to check your answer:
  ::所以,这个制度的解决方案是(11, -3)。 不要忘了检查你的答案:

  $$\begin{align*}2(11)+3(\text{-}3) &= 22-9=13\\ 11+5(\text{-}3) &= 11-15= \text{-}4 \end{align*}$$

  3.     

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  $$\begin{align*}4x+3y &=4\\ 6x-2y &= 19\end{align*}$$
  ::4x+3y=46x-2yy=19

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  In this case, none of the variables have a coefficient of 1. So  l et’s solve for the $\begin{align*}x\end{align*}$ in equation 1:
  ::在此情况下,没有一个变量的系数为1。 所以让我们解决方程式1中的 x 问题:

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  $$\begin{align*}4x &= \text{-}3y +4\\ x &= \text{-} \frac{3}{4}y+1\end{align*}$$
  ::4x=-3y+4x=-34y+1

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  Now this expression can be used to replace the $\begin{align*}x\end{align*}$ in the other equation and solve for $\begin{align*}y:\end{align*}$
  ::此表达式可用于替换其他方程式中的 x, 并解决 y:

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  $$\begin{align*}6 \left(\text{-} \frac{3}{4}y+1 \right)-2y &= 19\\ \text{-} \frac{18}{4}y+6-2y &= 19\\ \text{-} \frac{9}{2}y- \frac{4}{2}y &= 13\\ \text{-} \frac{13}{2}y &= 13\\ \left(\text{-}\frac{2}{13} \right)\left( \text{-} \frac{13}{2} \right)y &= 13\left( \text{-} \frac{2}{13} \right)\\ y &= \text{-}2\end{align*}$$
  ::6(-34y+1)-2y=19-184y+6-2y=19-92y-42y=13-132y=13(-213)(-132)(-132)y=13(-213)y=2

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  Now that we have found $\begin{align*}y\end{align*}$ , we can use this value in our expression find $\begin{align*}x:\end{align*}$
  ::既然我们找到了y, 我们可以在表达式查找 x 中使用这个值 :

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  $$\begin{align*}x &= \left(- \frac{3}{4} \right)(-2)+1\\ x &= \frac{6}{4}+1 \\ x &= \frac{3}{2}+\frac{2}{2}\\ x &= \frac{5}{2}\end{align*}$$
  ::x=(- 34)(-2)+1x=64+1x=32+22x=52

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  So, the solution is $\begin{align*}\left( \frac{5}{2}, \text{-}2 \right)\end{align*}$ . Check your answer:
  ::所以,答案是 (52,-2) 。请检查您的答案 :

  $$\begin{align*}4\left(\frac{5}{2}\right)+3(\text{-}2) &= 10-6=4\\ 6\left(\frac{5}{2}\right)-2(\text{-}2) &= 15+4 =19 \end{align*}$$

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to find the number of ounces of each of two concentrations of solution Rex and Carl need to make a particular mixture.
  ::早些时候,你被要求 找到两个浓度的溶液 每一盎司的数量 雷克斯和卡尔需要制造 一种特定的混合物。

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  Let’s try to make the word problem easier by organizing the  information into a “picture” equation as shown below:
  ::让我们把信息组织成下文所示的“图象”等式,

  

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  In this picture, we can see that they will be mixing $\begin{align*}x\end{align*}$ ounces of the 20% solution with $\begin{align*}y\end{align*}$ ounces of the 80% solution to get 12 ounces of the 60% solution. The two equations are thus:
  ::在这张图中,我们可以看到它们将把20%的溶液的x盎司与80%的溶液的y盎司混合,以获得60%的溶液的12盎司。

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  $$\begin{align*}0.2x+0.8y &= 0.6(12)\\ x+y &= 12\end{align*}$$
  ::0.2x+0.8.8y=0.6(12)x+y=12

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  Now we can solve the system using substitution. Solve for $\begin{align*}y\end{align*}$ in the second equation to get  $\begin{align*}y = 12-x.\end{align*}$
  ::现在我们可以用替换解决系统。 在第二个方程式中为 y 解决 y , 以获得 Y= 12 - x 。

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  Now, substitute and solve in the first equation:
  ::现在,第一个方程式的替代和解答:

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  $$\begin{align*}0.2x+0.8(12-x) &= 0.6(12)\\ 0.2x+9.6-0.8x &= 7.2\\ \text{-}0.6x &= \text{-}2.4\\ x &= 4\end{align*}$$
  ::0ux+0.8(12-x)=0.6(12)0.02x+9.6-0.8x=7.2-0.6x=-2.4x=4

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

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  $$\begin{align*}y &= 12-x\\ y &= 12-4\\ y &= 8\end{align*}$$
  ::y=12-xy=12-4y=8

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  Therefore, Rex and Carl need 4 ounces of the 20% saline solution and 8 ounces of the 80% saline solution to make the correct mixture.
  ::因此,Rex和Carl需要 20%盐碱溶液的4盎司 80%盐溶液的8盎司 才能做出正确的混合物

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  Solve the following systems using the substitution method.
  ::使用替代方法解决以下系统。

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  Example 2
  ::例2

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  $$\begin{align*}3x+4y &= \text{-}13\\ x &=\text{-}2y-9\end{align*}$$
  ::3x+4y=13x=-2y-9

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  In this problem, the second equation is already solved for $\begin{align*}x,\end{align*}$ so we can use that in the first equation to find $\begin{align*}y\end{align*}$ :
  ::在这个问题中,第二个方程式已经为 x 解决了, 所以我们可以在第一个方程式中使用它来找到 y:

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  $$\begin{align*}3(\text{-}2y-9)+4y &= \text{-}13\\ \text{-}6y-27+4y &= \text{-}13\\ \text{-}2y-27 &= \text{-}13\\ \text{-}2y &= 14\\ y &= \text{-}7\end{align*}$$
  ::3(3-2y-9)+4y=-13-6y-27+4y=-13-2y-27=-13-2y-y=-13-2y=-13-2y=14y=-7)

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

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  $$\begin{align*}x &= \text{-}2(\text{-}7)-9\\ x &= 14-9\\ x &= 5\end{align*}$$
  ::x=-2(-7)-9x=14-9x=5

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  Therefore the solution is (5, -7).
  ::因此,解决办法是(5,7) 。

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  Example 3
  ::例3

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  $$\begin{align*}\text{-}2x-5y &= \text{-}39\\ x+3y &=24\end{align*}$$
  ::-2x--5y=-39x+3y=24

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  This time the $\begin{align*}x\end{align*}$ in the second equation is the easiest variable to isolate: $\begin{align*}x=-3y+24.\end{align*}$  Let’s use this in the first expression to find $\begin{align*}y\end{align*}$ :
  ::这一次,第二个方程式中的 x 是最容易分离的变量 : x3y+24。 让我们在第一个表达式中使用它来查找 y :

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  $$\begin{align*}\text{-}2(\text{-}3y+24)-5y &= \text{-}39\\ 6y-48-5y &= \text{-}39\\ y-48 &= \text{-}39\\ y &=9\end{align*}$$
  :2-3y+24)-5y=-396y-48-5y=-39y-48=-39y=-39y=9

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

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  $$\begin{align*}x &= \text{-}3(9)+24\\ x &= \text{-}27+24\\ x &= \text{-}3\end{align*}$$
  ::x=-3(9)+24x=-27+24x=-3

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  Therefore the solution is (-3, 9).
  ::因此,解决办法是(-3,9)。

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  Example 4
  ::例4

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  $$\begin{align*}y &= \frac{1}{2}x-21\\ y &= \text{-}2x+9\end{align*}$$
  ::y=12x-21y=-2x+9

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  In this case, both equations are equal to $\begin{align*}y\end{align*}$ . Since $\begin{align*}y=y\end{align*}$ , by the Reflexive Property of Equality, we can set the right hand sides of the equations be equal too. This is still a substitution problem; it just looks a little different.
  ::在这种情况下,两个方程式都与y等同。y=y,根据“平等弹性属性 ” , 我们可以将方程式的右侧也设置为相等。 这仍然是一个替代问题; 它看起来有点不同了 。

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  $$\begin{align*}\frac{1}{2}x-21 &= \text{-}2x+9\\ 2 \left( \frac{1}{2}x-21 \right) &= 2(\text{-}2x+9)\\ x-42 &= \text{-}4x+18\\ 5x &= 60\\ x &= 12\end{align*}$$
  ::12-21=-2-2x+92(12x-21)=2(-2x+9)x-42=-4x+185x=60x=12

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

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  $$\begin{align*}y &= \frac{1}{2}(12)-21 \qquad \quad \quad y = \text{-}2(12)+9\\ y &= 6-21 \qquad \quad \text{or} \qquad y = \text{-}24+9\\ y &= \text{-}15 \qquad \qquad \qquad \quad y = \text{-}15\end{align*}$$
  ::y=12(12)-21y=-2(12)+9y=6-21y=24+9y=15

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  Therefore, the solution is (12, -15).
  ::因此,解决办法是(12,15)。

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  Review
  ::回顾

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  Solve the following systems using substitution. Remember to check your answers.
  ::使用替代解决以下系统。 记住检查您的答案 。

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  1. .   $\begin{align*}\begin{cases} x+3y= \text{-}1\\ 2x+9y= 7 \end{cases}\end{align*}$
     ::。 {x+3y=-12x+9y=7

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  2. .a $\begin{align*}\ \\ \begin{cases} 5x+2y= 0\\ y= x-7 \end{cases}\\ \ \\\end{align*}$  
     ::a. a. {5x+2y=0y=x-7

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  3. . $\begin{align*}\begin{cases} 5x+2y= 0\\ y= x-7 \end{cases}\end{align*}$  
     ::. 5x+2y=0y=x-7

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  4. . $\begin{align*}\ \\ \begin{cases} 2x-5y= 21\\ x= \text{-}6y +2 \end{cases}\\ \ \\\end{align*}$
     ::。 {2x-5y=21x=-6y+2

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  5. . $\begin{align*}\begin{cases} y= x+3\\ y= 2x-1 \end{cases}\end{align*}$  
     ::. y=x+3y=2x-1

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  6. . $\begin{align*}\ \\ \begin{cases}x+6y= 1\\ \text{-}2x-11y= \text{-}4 \end{cases}\\ \ \\\end{align*}$  
     ::。 {x+6y=1-2x-11y=4

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  7. .   $\begin{align*}\begin{cases}2x+y= 18\\ \text{-}3x+11y= \text{-}27 \end{cases}\end{align*}$
     ::. {2x+y=18-3x+11y=27

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  8. . $\begin{align*}\ \\ \begin{cases}2x+3y= 5\\ 5x+7y= 8 \end{cases}\\ \ \\\end{align*}$  
     ::。 {2x+3y=55x+7y=8

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  9. . $\begin{align*}\begin{cases}\text{-}7x+2y= 9\\ 5x-3y= 3 \end{cases}\end{align*}$  
     ::{-7x+2y=95x-3y=3

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  10. . $\begin{align*}\ \\ \begin{cases}2x-6y= \text{-}16\\ \text{-}6x+10y= 8 \end{cases}\\ \ \\\end{align*}$  
      ::。 {2x-6y=-16-6x+10y=8

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  11. . $\begin{align*}\begin{cases}2x-3y= \text{-}3\\ 8x+6y= 12 \end{cases}\end{align*}$  
      ::. {2x--3y=- 38x+6y=12

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  12. . $\begin{align*}\ \\ \begin{cases}5x+y= \text{-}3\\ y= 15x+9\end{cases}\\ \ \\\end{align*}$  
      ::。 {5x+y=-3y=15x+9

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  Set up and solve a system of linear equations to answer each of the following word problems.
  ::设置并解决一个线性方程式系统, 以回答以下每个字的问题 。

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  13. Alicia and Sarah are at the supermarket. Alicia wants to get peanuts from the bulk food bins and Sarah wants to get almonds. The almonds cost $6.50 per pound and the peanuts cost $3.50 per pound. Together they buy 1.5 pounds of nuts. If the total cost is $6.75, how much did each girl get? Set up a system to solve using substitution.
      ::Alicia和Sarah在超市。Alicia想从散装食品中买花生,Sarah想买杏仁。杏仁每磅6.50美元,花生每磅3.50美元。他们一起买1.5磅坚果。如果总成本是6.75美元,每个女孩能得到多少?建立一个系统,用替代方法解决问题。
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  14. Marcus goes to the department store to buy some new clothes. He sees a sale on t-shirts ($5.25) and shorts ($7.50). Marcus buys seven items and his total, before sales tax, is $43.50. How many of each item did he buy?
      ::马库斯去百货商店买些新衣服。他看到一件T恤(5.25美元)和短裤(7.50美元)的销售。马库斯买了七件,在销售税之前,他的总额是43.50美元。他每件都买了多少件?
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  15. Jillian is selling tickets for the school play. Student tickets are $3 and adult tickets are $5. If 830 people buy tickets and the total revenue is $3104, how many students attended the play?
      ::吉莉安在卖学校游戏的票。学生票是3美元,成人票是5美元。如果830人买票,总收入是3104美元,有多少学生参加了这场比赛?

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  Review (Answers)
  ::回顾(答复)

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  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 。