9.13 线性确认系统
章节大纲
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Michelle has filled her piggy bank with nickels and quarters. When she empties her bank she has a total of 135 coins. Michelle counts her money and is pleased to know she has a total of $18.75. How many of each coin did Michelle have in her bank?
::米歇尔已经用镍币和硬币填满了小猪银行。 当她把硬币和硬币空出时,她总共拥有135枚硬币。 米歇尔清点了自己的钱,并很高兴知道她总共拥有18.75美元。 米歇尔在她的银行里存了多少一枚硬币?In this concept, you will learn to recognize linear systems of equations.
::在这个概念中,你会学会认识线性方程系统。Linear System
::线性系统A linear equation with two variables can be written in standard form such that and are coefficients of the variables ‘ x ’ and ‘ y ’ and is a constant . The equation is linear since the variables ‘ x ’ and ‘ y ’ are only to the first power.
::包含两个变量的线性方程式可以以标准格式写成,使 A和 B 是变量 'x ' 和`y ' 的系数,而 C 是一个不变的系数。 这个方程式是线性方程式,因为变量 'x ' 和`y ' 只属于第一力量。A x + B y = C
::Ax+By=C 轴+By=CA linear system of two equations with two variables is any system that can be written in the form:
::包含两个变量的两个方程式的线性系统为以窗体写成的任何系统:A 1 x + B 1 y = C 1 A 2 x + B 2 y = C 2
::A1x+B1y=C1A2x+B2y=C2The variables ‘ x ’ and ‘ y ’ must be only in the numerator and there can be no products variables in the equations. An example of a linear system of equations with two variables is shown below:x + y = 12 x − y = 4
::x+y=12x-y=4 x+y=4Remember to solve an equation with two variables means to find an ordered pair that will make the statement of equality true.
::记住要用两个变量来解答一个方程式, 找到一个定购对配(x,y), 使平等声明真实化 。Let’s look at the first equation.
::让我们来看看第一个方程式。
::x+y=12 x+y=12The solution for the equation are any two numbers that have a sum of 12. There are many solutions for this equation. Some examples are
but there are an infinite number of ordered pairs that will make the statement true.
::等式的解决方案是具有12个总和的任何两个数字。 这个等式有许多解决方案。 一些例子有( 12,0, (15)-3, (9, 3, 3, (25)- 13), (8, 4), 但有无限数量的定购配对可以使声明成为事实。Let’s look at the second equation.
::让我们来看看第二个方程式。The solution for the equation are any two numbers that have a difference of 4. There are also many solutions for this equation. Some examples are but there are an infinite number of ordered pairs that will make the statement true.
::等式的解决方案是任何两个数字,两个数字的差别是4.,这个等式的解决方案也很多。有些例子有(12,8,(15,11),(9,5),(8,4),(2)-2,但数量无限的定购配对将使声明成为事实。The solution to a system of linear equations is a value for ‘ x ’ and a value for ‘ ’ that, when substituted into the equations, makes both equations a true statement of equality.
::线性方程式的解决方案是`x ' 值和`y ' 值,当被方程式取代时,使两个方程式成为真正的平等声明。Let’s test some of the ordered pairs from above into the equations.
::让我们从上面测试一些定购的对子到方程。Substitute the ordered pairs into the equation to see if it will make the statement true.
::将定购的对子换成方程式 看看它是否能使声明成为事实( 9 , 3 )
First, begin with the first equation.
::首先,从第一个方程开始x + y = 12
::x+y=12 x+y=12Next, substitute and into the equation.
::接下来,在方程中替换 x=9 和 y=3 。x + y = 12 9 + 3 = 12
::x+y=129+3=12Then, simplify the left side of the equation.
::然后,简化方程的左侧。9 + 3 = 12 12 = 12
Both sides of the equation are equal.
::等式的两面是平等的。The ordered pair
is a solution for the first equation.
::定购对(9,3)是第一个方程的解决方案。( 9 , 3 )
Now, test the ordered pair in the second equation.
::现在,测试第二个方程的定购配对x − y = 4
::x- y=4Next, substitute
and into the equation.
::接下来,在方程中替换 x=9 和 y=3 。x − y = 4 9 − 3 = 4
::x-y=49-3=4Then, simplify the left side of the equation.
::然后,简化方程的左侧。9 − 3 = 4 6 = 4 6 ≠ 4
Both sides of the equation are not equal.
::等式的两面不相等。The ordered pair
is NOT a solution for the second equation.
::定购对(9,3)不是第二个方程式的解决方案。Substitute the ordered pairs into the equation to see if it will make the statement true.
::将定购的对子换成方程式 看看它是否能使声明成为事实First, begin with the first equation.
::首先,从第一个方程开始x + y = 12
::x+y=12 x+y=12Next, substitute
and into the equation.
::接下来,在方程中替换 x=8和y=4。x + y = 12 8 + 4 = 12
::x+y=128+4=12Then, simplify the left side of the equation.
::然后,简化方程的左侧。8 + 4 = 12 12 = 12
Both sides of the equation are equal.
::等式的两面是平等的。The ordered pair
is a solution for the first equation.
::订购的对(8,4)是第一个方程的解决方案。( 8 , 4 )
Now, test the ordered pair in the second equation.
::现在,测试第二个方程的定购配对x − y = 4
::x- y=4Next, substitute
and into the equation.
::接下来,在方程中替换 x=8和y=4。x − y = 4 8 − 4 = 4
::x-y=48-4=4Then, simplify the left side of the equation.
::然后,简化方程的左侧。8 − 4 = 4 4 = 4
Both sides of the equation are equal.
::等式的两面是平等的。The ordered pair
is a solution for the second equation.
::订货对(8,4)是第二个方程的解决方案。The ordered pair
is the solution for the system of linear equations since and makes both equations true.
::定单对(8,4)是线性方程系统的解决方案,因为 x=8 和y=4 使两个方程都属实。x + y = 12 x − y = 4
::x+y=12x-y=4 x+y=4The solution can be written as ( x y ) = ( 8 4 ) .
and or as
::溶液可以以 x=8 和 y=4 或 (xy) =(84) 的形式写入 。Let’s look at another system of linear equations.
::让我们看看另一个线性方程式系统。x + y = − 4 2 x + 2 y = − 8
::xy+%% 4x+2y+% 8Some ordered pairs that have a sum of
and that satisfy the first equation are and . Do these ordered pairs satisfy the second equation?
::一些定购的配对,其总和为-4,符合第一个等式的配对是(-5,1,(-10,6)和(7,-11)。这些定购的配对是否满足第二个等式?( − 5 , 1 )
First, write down the second equation.
::首先,写下第二个方程2 x + 2 y = − 8
::2+2y8Next, substitute
and into the equation.
::下一步,在方程中替换 x5 和 y=1。2 x + 2 y = − 8 2 ( − 5 ) + 2 ( 1 ) = − 8
::2+2y82(-5)+2(1)8Then, simplify the left side of the equation.
::然后,简化方程的左侧。2 ( − 5 ) + 2 ( 1 ) = − 8 − 10 + 2 = − 8 − 8 = − 8
Both sides of the equation are equal.
::等式的两面是平等的。The ordered pair
is a solution for the second equation.
::定购对(-5,1)是第二个方程的解决方案。( − 10 , 6 )
First, write down the second equation.
::首先,写下第二个方程2 x + 2 y = − 8
::2+2y8Next, substitute
and into the equation.
::下一步,在方程中替换 x10和y=6。2 x + 2 y = − 8 2 ( − 10 ) + 2 ( 6 ) = − 8
::2+2y82(- 10)+2(6)8Then, simplify the left side of the equation.
::然后,简化方程的左侧。2 ( − 10 ) + 2 ( 6 ) = − 8 − 20 + 12 = − 8 − 8 = − 8
Both sides of the equation are equal.
::等式的两面是平等的。The ordered pair
is a solution for the second equation.
::定购对(-10,6)是第二个方程的解决方案。( 7 , − 11 )
First, write down the second equation.
::首先,写下第二个方程2 x + 2 y = − 8
::2+2y8Next, substitute
and into the equation.
::接下来,在方程中替换 x=7 和 y11 。2 x + 2 y = − 8 2 ( 7 ) + 2 ( − 11 ) = − 8
::2+2y82(7)+2(-11)8Then, simplify the left side of the equation.
::然后,简化方程的左侧。2 ( 7 ) + 2 ( − 11 ) = − 8 14 − 22 = − 8 − 8 = − 8
Both sides of the equation are equal.
::等式的两面是平等的。The ordered pair
is a solution for the second equation.
::定购对(7,-11)是第二个方程式的解决方案。There are many ordered pairs that solve both equations in the system. In fact, there is an infinite number of solutions that solve both linear equations in this system. The second equation in the system is a multiple of the first equation.
::许多定购对子解决了系统中的两个方程式。 事实上,在这个系统中,有无限数量的解决方案解决了两个线性方程式。 这个系统中的第二个方程式是第一个方程式的倍数。x + y = − 4 2 ( x + y = − 4 ) 2 x + 2 y = − 8
::x+y @% @% 42( x+y @% 4) 2x+2y @% 8All systems of linear equations such that one equation is a multiple of the other will have an infinite number of solutions.
::所有线性方程式系统,如一个方程式是另一个方程式的倍数,都将有无限数量的解决方案。Let’s look at another system of linear equations.
::让我们看看另一个线性方程式系统。x + y = 2 x + y = 5
::x+y=2x+y=5 x+y=5The ordered pair that would satisfy both of the equations in this system would have to add to give 2 and also add to give 5. There is no such ordered pair that would make both equations true.
::符合此系统中两种方程式的定购对子必须加上2,加上5,没有这种定购对子能够使两种方程式都真实。Rewrite both equations in slope-intercept form .
::以斜坡界面形式重写两个方程式。First, subtract ‘ x ’ from both sides of the equations to express the equations in the form .
::首先,从方程的两侧减去 'x ' ,以表示y=mx+b形式的方程。x + y = 2 x − x + y = 2 − x y = 2 − x
::x+y=2x-x+y=2-xy=2-xy=2-xy=2-xx + y = 5 x − x + y = 5 − x y = 5 − x
::x+y=5x-x+y=5-xy=5-xy=5-xThen, rewrite the right side of the equation to match the slope-intercept form
.
::然后重写方程的右侧, 以匹配 y=mx+b 的斜度界面 。y = m x + b y = 2 − x y = − x + 2
::y=mx+by=2 -xyx+2y = m x + b y = 5 − x y = − x + 5
::y=mx+by=5 -xy=xx+5 y=mx+by=5 -xy=xx+5Notice that both equations have the same of intercepts , and .
but different -
::注意两个方程式的 m1 相同, 但y interviews, b=2 和 b=5 不同 。Lines having the same slope are parallel and parallel lines do not have any ordered pairs in common. Therefore , linear equations that have the same slope and different -intercepts will have no solution.
::具有相同斜度的线条是平行的,而平行线条没有一对相同的定单对。 因此,具有相同斜度和不同 Y 截面的线性方程式将无法解决问题。Examples
::实例Example 1
::例1Earlier, you were given a problem about Michelle and her piggy bank. She needs to figure out how many nickels and how many quarters she had in her bank. How can Michelle do this?
::早些时候,你遇到了米歇尔和她的小猪银行的问题。她需要弄清楚她银行里有多少镍币和多少食堂。米歇尔怎么能这样做呢?She can write a system of linear equations to model the information given and find the solution for the system of equations .
::她可以写出一个线性方程式系统,以模拟所提供的信息,并为方程式系统找到解决办法。First, write down the information given in the problem.
::首先,写下在问题中提供的信息。The number of nickels plus the number of quarters equals 135 coins. The value of the nickels plus the value of the quarters equals $18.75.
::镍的数量加上硬币的数量等于135硬币,镍的价值加上硬币的价值等于18.75美元。Next, write the system of equations to model the given information.
::下一步,写入方程式系统以模拟给定信息。Let
and let .
::让我们 x = 镍数, y = 季度数 。x + y = 135 0.05 x + 0.25 y = 18.75
::x+y=1350.05x+0.25y=18.75Next, write down ordered pairs that have a sum of 135.
::接下来,写下定购的一对, 总共135美元。( 65 , 70 ) , ( 60 , 75 ) , ( 70 , 65 ) and ( 75 , 60 )
:65,70,(60,75),(70,65)和(75,60)
All of these ordered pairs will satisfy the first equation. Which ordered pair will make the second equation true?
::所有这些定购的配对将满足第一个方程。哪个定购的配对将使第二个方程成为真实?First, write down the second equation.
::首先,写下第二个方程0.05 x + 0.25 y = 18.75
::0.05x+0.25y=18.75Next, substitute
and into the equation.
::接下来,在方程中替换 x=65和y=70。0.05 x + 0.25 y = 18.75 0.05 ( 65 ) + 0.25 ( 70 ) = 18.75
::0.05x+0.25y=18.750.05(65)+0.25(70)=18.75Then, simplify the left side of the equation.
::然后,简化方程的左侧。0.05 ( 65 ) + 0.25 ( 70 ) = 18.75 3.25 + 17.50 = 18.75 20.75 ≠ 18.75
Both sides of the equation are NOT equal.
::等式的两面是不平等的。The ordered pair
is NOT a solution for the second equation.
::定购对(65,70)不是第二个方程的解决方案。First, write down the second equation.
::首先,写下第二个方程0.05 x + 0.25 y = 18.75
::0.05x+0.25y=18.75Next, substitute
and into the equation.
::接下来,在方程中替换 x=60和y=75。0.05 x + 0.25 y = 18.75 0.05 ( 60 ) + 0.25 ( 75 ) = 18.75
::0.05x+0.25y=18.750.05(60)+0.25(75)=18.75Then, simplify the left side of the equation.
::然后,简化方程的左侧。0.05 ( 60 ) + 0.25 ( 75 ) = 18.75 3.00 + 18.75 = 18.75 21.75 ≠ 18.75
Both sides of the equation are NOT equal.
::等式的两面是不平等的。The ordered pair
is NOT a solution for the second equation.
::定购对(60,75)不是第二个方程的解决方案。First, write down the second equation.
::首先,写下第二个方程0.05 x + 0.25 y = 18.75
::0.05x+0.25y=18.75Next, substitute
and into the equation.
::接下来,在方程中替换 x=70 和 y=65 。0.05 x + 0.25 y = 18.75 0.05 ( 70 ) + 0.25 ( 65 ) = 18.75
::0.05x+0.25y=18.750.05(70)+0.25(65)=18.75Then, simplify the left side of the equation.
::然后,简化方程的左侧。0.05 ( 70 ) + 0.25 ( 65 ) = 18.75 3.50 + 16.25 = 18.75 19.75 ≠ 18.75
Both sides of the equation are NOT equal.
::等式的两面是不平等的。The ordered pair
is NOT a solution for the second equation.
::定购对(70,65)不是第二个方程的解决方案。First, write down the second equation.
::首先,写下第二个方程0.05 x + 0.25 y = 18.75
::0.05x+0.25y=18.75Next, substitute
and into the equation.
::接下来,在方程中替换 x=75和y=60。0.05 x + 0.25 y = 18.75 0.05 ( 75 ) + 0.25 ( 60 ) = 18.75
::0.05x+0.25y=18.750.05(75)+0.25(60)=18.75Then, simplify the left side of the equation.
::然后,简化方程的左侧。0.05 ( 75 ) + 0.25 ( 60 ) = 18.75 3.75 + 15.00 = 18.75 18.75 = 18.75
Both sides of the equation are equal.
::等式的两面是平等的。The ordered pair
is a solution for the second equation.
::定购对(75,60)是第二个方程的解决方案。The solution is ( x y ) = ( 75 60 ) .
::解决方案是 (xy) = (7560) 。Michelle had 75 nickels and 60 quarters in her piggy bank.
::Michelle在她的小猪银行里有75分和60分Example 2
::例2Which ordered pair makes both equations true?
::哪一对定购的方程式使两个方程式都属实?x + y = 8 4 x − y = − 3
::x+y=84x-y3First, substitute
and into both equations of the system.
::首先,将 x=2 和 y=6 替换为系统的两个方程。x + y = 8 2 + 6 = 8
::x+y=82+6=8 x+y=82+6=84 x − y = − 3 4 ( 2 ) − 6 = − 3
::4 - y34(2) - 63Next, simplify the left side of each of the equations.
::接下来,简化每个方程的左侧。2 + 6 = 8 8 = 8
4 ( 2 ) − 6 = − 3 8 − 6 = − 3 2 ≠ − 3
The ordered pair
makes the first equation true but it does NOT make the second equation true.
::定购的对(2,6)使第一个方程成为真实,但不能使第二个方程成为真实。First, substitute
and into both equations of the system.
::首先,在系统的两个方程中替换 x=3 和y=15。x + y = 8 3 + 15 = 18
::x+y=83+15=18 x+y=83+15=184 x − y = − 3 4 ( 3 ) − 15 = − 3
::4 - y 34 (3) - 15 3Next, simplify the left side of each of the equations.
::接下来,简化每个方程的左侧。3 + 15 = 8 18 ≠ 8
4 ( 3 ) − 15 = − 3 12 − 15 = − 3 − 3 = − 3
The ordered pair ( 3 , 15 ) does NOT make the first equation true but it does make the second equation true.
::订购的对(3,15)并不使第一个方程成为真实,但它确实使第二个方程成为真实。First, substitute
and into both equations of the system.
::首先,在系统的两个方程中替换 x=4和y=4。x + y = 8 4 + 4 = 8
::x+y=84+4=84 x − y = − 3 4 ( 4 ) − 4 = − 3
::4 - y34(4) - 43Next, simplify the left side of each of the equations.
::接下来,简化每个方程的左侧。4 + 4 = 8 8 = 8
4 ( 4 ) − 4 = − 3 16 − 4 = − 3 12 ≠ − 3
The ordered pair
makes the first equation true but it does NOT make the second equation true.
::定购的对(4,4)使第一个方程成为真实,但不能使第二个方程成为真实。First, substitute
and into both equations of the system.
::首先,在系统的两个方程中替换 x=1和y=7。x + y = 8 1 + 7 = 8
::x+y=81+7=84 x − y = − 3 4 ( 1 ) − 7 = − 3
::4 - y34(1) - 73Next, simplify the left side of each of the equations.
::接下来,简化每个方程的左侧。1 + 7 = 8 8 = 8
4 ( 1 ) − 7 = − 3 4 − 7 = − 3 − 3 = − 3
The ordered pair
makes the first equation true and it also makes the second equation true.
::定购一对(1,7)使第一个方程成为真实,也使第二个方程成为真实。The solution is ( x y ) = ( 1 7 ) .
::解决方案是 (xy) = (17) 。Example 3
::例3Will the following system of linear equations have one solution, no solution or an infinite number of solutions? Justify your answer.
::以下的线性方程式系统是否会有一个解决方案, 没有解决方案, 也没有无限数量的解决方案? 请说明答案是否合理 。y = 1 2 x − 3 y = 1 2 x + 2
::y= 12x- 3y= 12x+2First, write down what you know from the given equations.
::首先,写下您从给定方程式中知道的东西。The equations are written in the form
.
::方程式以y=mx+b的形式写成。Both equations have the same slope of
.
::两个方程式的斜坡均是12。The equations do not have the same
-intercepts.
::方程式没有相同的 Y 接口 。Next, state the number of solutions for the system of equations.
::其次,说明方程系统的解决办法数量。Lines that are parallel have no ordered pairs in common so there will be no solution.
::平行的线条没有一对一对成单,所以没有解决办法。Example 4
::例4Will the following system of linear equations have one solution, no solution or an infinite number of solutions? Justify your answer.
::以下的线性方程式系统是否会有一个解决方案, 没有解决方案, 也没有无限数量的解决方案? 请说明答案是否合理 。2 3 x − 3 4 y = − 1 8 x − 9 y = − 12
::23-34y18x-9y12First, write down what you know from the given equations.
::首先,写下您从给定方程式中知道的东西。The second equation is a multiple of the first equation.
::第二个方程式是第一个方程式的倍数。4 12 ( 2 3 x ) − 3 12 ( 3 4 y ) = 12 ( − 1 ) 4 ( 2 x ) − 3 ( 3 y ) = 12 ( − 1 ) 8 x − 9 y = − 12 8 x − 9 y = − 12
::412(23x)-312(34y)=12(-1)4(2x)-3(3y)=12(-1)8x-9y128x-9y12Next, state the number of solutions for the system of equations.
::其次,说明方程系统的解决办法数量。When one equation in a system of linear equations is a multiple of the other equation, there will be an infinite number of solutions for the system.
::当一个线性方程式系统中的一个方程式是另一个方程式的倍数时,这个系统就会有无限数量的解决办法。Example 5
::例5Is the ordered pair
the solution for the following system of linear equations?
::定购对(3,-2)是下列线性方程系统的解决办法吗?x − 3 y = 9 3 x + y = 7
::x-3y=93x+y=7 x-3y=93x+y=7First, substitute
and into both equations of the system.
::首先,在系统的两个方程式中替换 x=3 和 y2。x − 3 y = 9 3 − 3 ( − 2 ) = 9
::x-3y=93-3(-2)=93 x + y = 7 3 ( 3 ) + ( − 2 ) = 7
::3x+y=73(3)+(-2)=7Next, simplify the left side of each of the equations.
::接下来,简化每个方程的左侧。3 − 3 ( − 2 ) = 9 3 + 6 = 9 9 = 9
3 ( 3 ) + ( − 2 ) = 7 9 − 2 = 7 7 = 7
The ordered pair
makes the first equation true and it also makes the second equation true.
::订购的对(3,-2)使第一个方程成为真实,也使第二个方程成为真实。The solution is ( x y ) = ( 3 − 2 ) .
::解决方案是 (xy) = (3-2) 。Review
::回顾Figure out which pair is a solution for each given system.
::找出哪种配对是每个特定系统的解决办法。1. Which ordered pair is a solution of the following system?
::1. 哪种定购对是以下系统的解决办法?
::x-3y=93x+y=7 x-3y=93x+y=72. Which ordered pair is a solution of the following system?
::2. 哪种定购配对是以下系统的解决办法?y = 3 x − 7 5 x − 3 y = 13
::y=3x- 75x- 3y=13- ( 4 , 7 )
Determine whether each system has infinite solutions or no solutions.
::确定每个系统是否有无限的解决方案或没有解决方案。3.
::3. x+y=10yx+104.
::4. 3x-6y24x-2y85.
::5. 34x=23y-19x=8y-126.
::6. y=3x-5y=3x-27.
::7. y=12x+3y=12x-2Answer each question true or false.
::回答每个问题是否真实或虚假。8. Parallel lines have the same slope.
::8. 平行线的斜坡相同。9. A linear system of equations cannot be graphed on the coordinate plane.
::9. 坐标平面上无法绘制线性方程图。10. Parallel lines have infinite solutions.
::10. 平行线有无限的解决办法。11. Perpendicular lines have one solution.
::11. 直直线有一个解决办法。12. Lines with an infinite number of solutions are not parallel.
::12. 有无数解决办法的线条并不平行。13. Some linear systems do not have a solution.
::13. 有些线性系统没有解决办法。14. To solve a linear system, you must have a value for x and .
::14. 要解决线性系统,您必须有 x 和 y 的值。15. An ordered pair is never a solution for a linear system.
::15. 定购一对从来就不是线性系统的解决办法。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 。Resources
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