9.14 使用交叉重叠解决合理等同

Section outline

• Select section General

  Collapse Expand

  General

  Collapse all Expand all

  • Select activity 新闻通告

    

    新闻通告 Forum
• Select section 使用交叉乘法解析理衡

  Collapse Expand

  使用交叉乘法解析理衡

  播放段落

  A scale model of a racecar is in the ratio of 1: x to the real racecar. The length of the model is $\begin{array}{r}2x-21\end{array}$ units, and the length of the real racecar is $\begin{array}{r}{x}^2\end{array}$ units. What is the value of x ?
  ::赛车规模模型的比重为 1:x 与 实际赛车的比重。 模型的长度为 2x- 21 个单位,而实际赛车的长度为 x2 个单位。 x 值是多少?

  播放段落

  Solving Rational Equations Using Cross Multiplication
  ::使用交叉乘法解决合理等式

  播放段落

  A rational equation is an equation where there are rational expressions on both sides of the equal sign. One way to solve rational equations is to use cross- multiplication . Here is an example of a proportion that we can solve using cross-multiplication.
  ::理性方程式是一个等式, 它在等号的两侧都有合理的表达方式。 解答理性方程式的一个方法就是使用交叉乘法。 这是一个比例的示例, 我们可以使用交叉乘法来解决 。

  

  播放段落

  Let's u se cross multiplication to solve the following equations.
  ::让我们使用交叉乘法来解决以下方程。

  播放段落
  1. 播放段落

     $\begin{array}{r}\frac{x}{2x-3}=\frac{3x}{x+11}\end{array}$
     ::x2x-3=3xx+11

     
     ::x2x-3=3xx+11

  播放段落

  Cross-multiply and solve.
  ::交叉倍数和解答。

  

  播放段落

  Check your answers. It is possible to get extraneous solutions with rational expressions.
  ::检查您的答案。 可以用理性表达式获得不相干的解决办法 。

  $$\begin{array}{rlrl}\frac{0}{2\cdot 0-3}& =\frac{3\cdot 0}{0+11}& & \frac{4}{2\cdot 4-3}=\frac{3\cdot 4}{4+11}\\ \frac{0}{-3}& =\frac{0}{11}& & \frac{4}{5}=\frac{12}{15}\\ 0& =0& & \frac{4}{5}=\frac{4}{5}\end{array}$$

  播放段落
  2. $\begin{array}{r}\frac{x+1}{4}=\frac{3}{x-3}\end{array}$
     ::x+14=3x-3

  播放段落

  Cross-multiply and solve.
  ::交叉倍数和解答。

  播放段落

  $$\begin{array}{rl}\frac{x+1}{4}& =\frac{3}{x-3}\\ 12& =x^2-2x-3\\ 0& =x^2-2x-15\\ 0& =(x-5)(x+3)\\ x& =5and-3\end{array}$$

  
  ::x+14=3x-312=x2-2x-30=x2-30=x2-2x-150=(x-5)(x+5)(x+3)x=5和-3

  播放段落

  Check your answers.
  ::检查你的答案。

  播放段落

  $\begin{array}{r}\frac{5+1}{4}=\frac{3}{5-3}\to \frac{6}{4}=\frac{3}{2}\end{array}$ and
  ::5+14=35-364=32和

  $\begin{array}{r}\frac{-3+1}{4}=\frac{3}{-3-3}\to \frac{-2}{4}=\frac{3}{-6}\end{array}$

  播放段落
  3. $\begin{array}{r}\frac{x^2}{2x-5}=\frac{x+8}{2}\end{array}$
     ::x22x- 5=x+82

  播放段落

  Cross-multiply and solve.
  ::交叉倍数和解答。

  播放段落

  $$\begin{array}{rl}\frac{x^2}{2x-5}& =\frac{x+8}{2}\\ 2x^2+11x-40& =2x^2\\ 11x-40& =0\\ 11x& =40\\ x& =\frac{40}{11}\end{array}$$

  
  ::x22x- 5=x+822x2+11x- 40=2x221x- 40=011x=011x=40x=4011

  播放段落

  Check the answer: $\begin{array}{r}\frac{{\left(\frac{40}{11}\right)}^2}{\frac{80}{11}-5}=\frac{\frac{40}{11}+8}{2}\to \frac{1600}{121}÷\frac{25}{11}=\frac{128}{11}÷2\to \frac{64}{11}=\frac{128}{22}\end{array}$
  ::答: (4011) 28011- 5= 4011+82_ 1600121_ @ 2511=12811=2_ 6411=12822

  播放段落

  Examples
  ::实例

  播放段落

  Example 1
  ::例1

  播放段落

  Earlier, you were asked to find the value of x. 
  ::早些时候,你被要求找到x的值。

  播放段落

  We need to set up a rational equation and solve for x .
  ::我们需要设置一个理性的方程 并解决x。

  播放段落

  $\begin{array}{r}\frac{1}{x}=\frac{2x-21}{x^2}\end{array}$
  ::1x=2x-21x2

  播放段落

  Now cross-multiply.
  ::现在交叉倍增

  播放段落

  $$\begin{array}{r}{x}^2=x(2x-21)\\ x^2=2x^2-21x\\ 0=x^2-21x\\ 0=x(x-21)\\ x=0,21\end{array}$$

  
  ::x2=xx(2x-21xx2=2x2-21x0=x2-21x0=x2-21x0=xx(x-21x)x=0,21

  播放段落

  However, x is a ratio so it must be greater than 0. Therefore x equals 21 and the model is in the ratio 1:21 to the real racecar.
  ::然而,x是一个比率,因此它必须大于0。因此,x等于21,模型是实际赛车的1:21比1。

  播放段落

  Solve the following rational equations.
  ::解决以下合理的方程式。

  播放段落

  Example 2
  ::例2

  播放段落

  $\begin{array}{r}\frac{-x}{x-1}=\frac{x-8}{3}\end{array}$
  ::-xx-1=x-83

  播放段落

  
  

  $$\begin{array}{rl}\frac{-x}{x-1}& =\frac{x-8}{3}\\ x^2-9x+8& =-3x\\ x^2-6x+8& =0\\ (x-4)(x-2)& =0\\ x& =4and2\end{array}$$

  
  ::-xx-1=x-83x2-9x+8*3x2-6x8=0(x-4)(x-2)=0x=4和2

  播放段落

  $$\begin{array}{rlrl}\text{Check}:x=4\to \frac{-4}{4-1}& =\frac{4-8}{3}& & x=2\to \frac{-2}{2-1}=\frac{2-8}{3}\\ \frac{-4}{3}& =\frac{-4}{3}& & \frac{-2}{1}=\frac{-6}{3}\end{array}$$

  
  ::检查:x=444- 1=4-83x=222-1=2-83-43}43-2163

  播放段落

  Example 3
  ::例3

  播放段落

  $\begin{array}{r}\frac{x^2-1}{x+2}=\frac{2x-1}{2}\end{array}$
  ::x2 - 1x+2=2x- 12

  播放段落

  
  

  $$\begin{array}{rl}\frac{x^2-1}{x+2}& =\frac{2x-1}{2}\\ 2x^2+3x-2& =2x^2-2\\ 3x& =0\\ x& =0\end{array}$$

  
  ::x2 - 1x+2=2x - 122x2+3x-2=2x2 - 23x=0x=0

  播放段落

  $$\begin{array}{rl}\text{Check}:\frac{0^2-1}{0+2}& =\frac{2\left(0\right)-1}{2}\\ \frac{-1}{2}& =\frac{-1}{2}\end{array}$$

  
  ::检查:02- 10+2=2( 0)- 12- 12 @ 12

  播放段落

  Example 4
  ::例4

  播放段落

  $\begin{array}{r}\frac{9-x}{x^2}=\frac{4}{-3x}\end{array}$
  ::9-2x2=4-3x

  播放段落

  
  

  $$\begin{array}{rl}\frac{9-x}{x^2}& =\frac{4}{-3x}\\ 4x^2& =-27x+3x^2\\ x^2+27x& =0\\ x(x+27)& =0\\ x& =0and-27\end{array}$$

  
  ::9 -xx2=4 - 3x4x2\27x+3x2x2x2+27x=0xxxxxx(x+27=0x=0和-27

  播放段落

  $$\begin{array}{rlrl}\text{Check}:x=0\to \frac{9-0}{0^2}& =\frac{4}{-3\left(0\right)}& & x=-27\to \frac{9+27}{{\left(-27\right)}^2}=\frac{4}{-3\left(-27\right)}\\ und& =und& & \frac{36}{729}=\frac{4}{81}\\ & & & \frac{4}{81}=\frac{4}{81}\end{array}$$

  
  ::检查:x=09- 002=4- 3(0)x_ 27_ 9+27(- 272)=4-3(- 27und=und36729=481 481=481=481

  播放段落

  $\begin{array}{r}x=0\end{array}$ is not actually a solution because it is a vertical asymptote for each rational expression , if graphed. Because zero is not part of the domain , it cannot be a solution, and is extraneous.
  ::x=0 不是一个实际的解决方案, 因为它是每个理性表达式的垂直零点, 如果图形化的话。 因为零不是域的一部分, 它不能是一个解决方案, 而且是外在的 。

  播放段落

  Review
  ::回顾

  播放段落
  1. Is $\begin{array}{r}x=-2\end{array}$ a solution to $\begin{array}{r}\frac{x-1}{x-4}=\frac{x^2-1}{x+4}\end{array}$ ?
     ::x% 2 是 x- 1x-4 = x2 - 1x+4 的解决方案吗 ?

  播放段落

  Solve the following rational equations.
  ::解决以下合理的方程式。

  播放段落
  2. $\begin{array}{r}\frac{2x}{x+3}=\frac{8}{x}\end{array}$
     ::2xx+3=8x
  播放段落
  3. $\begin{array}{r}\frac{4}{x+1}=\frac{x+2}{3}\end{array}$
     ::4x+1=x+23 4x+1=x+23
  播放段落
  4. $\begin{array}{r}\frac{x^2}{x+2}=\frac{x+3}{2}\end{array}$
     ::x2x+2=x+32
  播放段落
  5. $\begin{array}{r}\frac{3x}{2x-1}=\frac{2x+1}{x}\end{array}$
     ::3x2x- 1=2x+1x
  播放段落
  6. $\begin{array}{r}\frac{x+2}{x-3}=\frac{x}{3x-2}\end{array}$
     ::x+2x- 3=x3x-2
  播放段落
  7. $\begin{array}{r}\frac{x+3}{-3}=\frac{2x+6}{x-3}\end{array}$
     ::x+3 - 3=2x+6x- 3
  播放段落
  8. $\begin{array}{r}\frac{2x+5}{x-1}=\frac{2}{x-4}\end{array}$
     ::2x+5x-1=2x-4
  播放段落
  9. $\begin{array}{r}\frac{6x-1}{4x^2}=\frac{3}{2x+5}\end{array}$
     ::6-14x2=32x+5
  播放段落
  10. $\begin{array}{r}\frac{5x^2+1}{10}=\frac{x^3-8}{2x}\end{array}$
      ::5x2+110=x3-82x
  播放段落
  11. $\begin{array}{r}\frac{x^2-4}{x+4}=\frac{2x-1}{3}\end{array}$
      ::x2 - 4x+4=2x- 13

  播放段落

  Determine the values of a that make each statement true. If there no values, write none.
  ::确定使每个语句真实的值。 如果没有数值, 写不出来 。

  播放段落
  12. $\begin{array}{r}\frac{1}{x-a}=\frac{x}{x+a}\end{array}$ , such that there is no solution.
      ::1xa=xx+a,因此没有解决办法。
  播放段落
  13. $\begin{array}{r}\frac{1}{x-a}=\frac{x}{x-a}\end{array}$ , such that there is no solution.
      ::1 -a=xx-a,因此没有解决办法。
  播放段落
  14. $\begin{array}{r}\frac{x-a}{x}=\frac{1}{x+a}\end{array}$ , such that there is one solution.
      ::x-ax=1x+a, 只有一个解决方案 。
  播放段落
  15. $\begin{array}{r}\frac{1}{x+a}=\frac{x}{x-a}\end{array}$ , such that there are two integer solutions.
      ::1x+a=xx-a,有两种整数解决办法。

  播放段落

  Review (Answers)
  ::回顾(答复)

  播放段落

  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 。