4.12 摘要:线性等同和不平等体系
章节大纲
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In this chapter, we learned about:
::在本章中,我们了解到:What Is a System of Equations And What Are the Possible Solutions
::何为等同体系? 何为可能的解决办法?-
A system of equations is a group of equations that we consider together at one time.
::方程式系统是一组我们同时一起考虑的方程式。 -
A solution to a system of equations is a point or a set of points that makes all equations true.
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If the system of equations has distinct graphs, then it is independent.
::如果方程式系统有不同的图形,那么它是独立的。 -
If the system of equation has graphs that coincide at all points, then it is dependent.
::如果方程式系统中的图表在所有点都吻合,则取决于该方程式。 -
A system that has no solution is inconsistent.
::一个没有解决办法的制度是不一致的。 -
Consistent systems are systems which have at least one solution.
::统一的系统是至少有一种解决办法的系统。 -
For systems of equations with two variables, we can determine these properties by graphing, but we can also compare these properties by comparing the slopes and the
y
-intercepts of the lines.
::对于具有两个变量的方程系统,我们可以通过图形化来确定这些属性,但我们也可以比较这些属性,比较斜坡和线的 Y 截面。
::方程式系统的解决办法是使所有方程式都真实的点或一组点。如果方程式系统有不同的图形,那么它是独立的。如果方程式系统有在所有点都一致的图形,那么它是独立的。没有解决方案的系统是不一致的。一致的系统是至少有一个解决方案的系统。对于具有两个变量的方程式系统,我们可以通过图形化来确定这些属性,但我们也可以比较这些属性,比较斜坡和线条的 Y 界面。 -
If the system of equations has distinct graphs, then it is independent.
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Three-dimensional space is made up of three intersecting number lines.
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Points in 3-space are of the form (
x
,
y
,
z
).
::3个空格中的点为窗体(x,y,z)。
::三维空间由三条交叉数字线组成。三维空间的点为形式(x,y,z)。 -
Points in 3-space are of the form (
x
,
y
,
z
).
-
Linear systems of three variables can have three possible solutions—one, none, or infinite.
::三个变数的线性系统可以有三个可能的解决办法——一个,一个,一个,或无限。
How To Solve Systems of Equations With Two Variables
::如何用两个变量解决等式系统-
To solve a system of equations by graphing, graph the lines together on the same set of axes. The solution(s) is (are) any points of intersection.
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It is best to use graphing when you may want to have a picture of the situation to analyze other aspects of the problem.
::最好在您想了解情况以分析问题其他方面时使用图表。
::要通过图形化来解析方程式系统, 请用图形化的方式将线条放在相同的轴上。 解决方案是任何交叉点 。 最好在您想要用图解图解来分析问题的其他方面时使用图解 。
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It is best to use graphing when you may want to have a picture of the situation to analyze other aspects of the problem.
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To solve a system of equations by substitution, isolate a variable in one of the equations and substitute into the other equation to find a value. Then, substitute the value into either equation.
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It is best to use substitution if one of the variables is isolated or has a coefficient of 1 or -1.
::如果一个变量是孤立的或系数为1或-1,则最好使用替代。
::要通过替代解决方程式系统,在其中一个方程式中分离一个变量,并替换到另一个方程式中以找到一个值。然后,将该值替换到任何一个方程式中。如果其中一个变量是孤立的或系数为1或-1,最好使用替代。 -
It is best to use substitution if one of the variables is isolated or has a coefficient of 1 or -1.
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To solve a system of equations by elimination by addition, identify additive inverses and then add the like terms in the equations. Then, substitute the found value into one of the equations to find the other value.
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It is best to use elimination by addition when the coefficients of one of the variables are additive inverses.
::如果其中一个变数的系数是复数反数,最好通过加法来消除这些变数。
::要通过添加去掉来解析方程式系统, 确定添加反函数, 然后在方程式中添加类似条件 。 然后, 将找到的值换成一个方程式以找到另一个值 。 当一个变量的系数是添加反函数时, 最好使用加法来消除该等方程式 。 -
It is best to use elimination by addition when the coefficients of one of the variables are additive inverses.
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To solve a system of equations by elimination by multiplication and addition, choose a variable to eliminate and find the LCM of the coefficients of that variable. Then, multiply both sides of the equations to make the coefficient of the variable you want to eliminate the LCM. Make sure these terms have opposite signs, so they are additive inverses of each other. Add the equations and solve for the variable if necessary. Once you have found one value, substitute into either equation to find the value of the variable you choose to eliminate.
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It is best to use elimination by multiplication and addition in the remaining cases.
::最好在其余情况下通过乘法和加法来消除这些疾病。
::要通过乘法和加法消除方程式系统,通过乘法和加法来解决方程式系统。 请选择一个变量来消除并找到该变量系数的 LCM。 然后, 将方程式的两边乘法来设定您想要消除 LCM 的变量的系数。 请确定这些术语有相反的符号, 因而它们是相加的反函数。 必要时, 添加方程式并解决变量 。 一旦您找到一个值, 则在两个方程式中取用一个方程式来找到您选择删除的变量的值 。 最好通过乘法和在剩余情况下加法来消除变量 。 -
It is best to use elimination by multiplication and addition in the remaining cases.
How To Solve a System of Inequalities
::如何解决不平等体系-
To graph a system of linear inequalities, graph each inequality on the same set of axes. The solution is the region where the shading for both inequalities overlaps.
::要绘制线性不平等体系图,请用图表显示同一组轴上的每一种不平等。 解决方案是两种不平等的阴影重叠的区域。
How To Solve a System of Equations With Three Variables
::如何解决三个变量的等式系统-
To solve a system of three linear equations, reduce the system to two equations with two unknowns either by elimination by multiplication and addition or by substitution. Then, solve as a system of two linear equations. Substitute to find the missing values.
::要解决由三个线性方程式组成的系统, 将系统缩小为两个未知方程式, 两个未知方程式, 要么通过乘法和加法来消除, 要么通过替换来消除。 然后, 以两个线性方程式的系统来解析。 替代以寻找缺失值 。
Looking Back, Looking Forward
::回顾,展望未来In this chapter, we learned techniques for considering two or three linear equations or inequalities at one time. However, we cannot model all situations with linear models. In the following chapters, we discuss other types of models, and in the next chapter, we lay the groundwork for that discussion.
::在本章中,我们学习了同时考虑两三个线性方程式或不平等的技巧,然而,我们不能用线性模式模拟所有情况。 在以下各章中,我们讨论了其他类型的模式,在下一章中,我们为这一讨论打下了基础。Chapter Review
::回顾章次审查 -
A system of equations is a group of equations that we consider together at one time.