这部分需要会什么？
- 根据直线关系区分唯一解、无解与无穷多解
- 通过消元求使解的个数变化的参数
- 检查系数为零等退化情形
核心概念与公式
对于 \(a_1x+b_1y=c_1\) 和 \(a_2x+b_2y=c_2\)，若 \(a_1b_2-a_2b_1\ne0\)，就有唯一解。若该值为零，应继续消元：得到假等式则无解；得到恒等式且剩余约束相容，则有无穷多解。
两个方程确实都表示直线时：相交对应唯一解，平行且不重合对应无解，重合对应无穷多解。判断重合时，变量系数和常数必须按同一个倍数变化。
做题时怎样使用
- 先检查方程是否能表示直线，留意两个变量系数同时为零的情况
- 用倍乘消元比较常数，避免对可能为零的系数直接作比
- 求出参数后代回原方程，确认得到的是平行、重合还是相交
容易混淆的地方
系数成比例只能说明没有唯一交点；常数是否同倍变化决定无解还是无穷多解。\(0x+0y=0\) 不代表一条直线。
用一个例题串起方法
知识点教学例题方程组 \(2x+3y=7\)，\(4x+ky=15\) 在什么 \(k\) 值下无解？
展开解题思路 ＋
把第一式乘 2 得 \(4x+6y=14\)。当 \(k=6\) 时，同一左边同时等于 14 和 15，矛盾，所以无解；当 \(k\ne6\) 时，可消元解出唯一的 \(y\) 和 \(x\)。答案为 6。
本知识点题库
先独立作答，再查看答案与解析。
方程组与方程解的个数基础
Which system of linear equations has no solution?
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方程组与方程解的个数挑战
\( nx+3t=-3x+4n \)
In the given equation, \( n \) and \( t \)are constants. The equation has no solution. What CANNOT be the value of \( n+t \)?
知序中文精讲
方程组与方程解的个数进阶
\( \frac{2}{5}x+\frac{7}{5}y=\frac{2}{7} \)
\( gx+ky=\frac{5}{2} \)
In the given system of equations, g and k are constants. The system has infinitely many solutions. What is the value of \( \frac{g}{k} \)?
知序中文精讲
方程组与方程解的个数基础
y = 6x + 18One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?
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方程组与方程解的个数进阶
\( 30x=1,800y-2,700 \)
One of the two equations in a system of linear equations is given. The system has no solution. Which equation could be the second equation in this system?
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方程组与方程解的个数进阶
\( \frac{4}{5}x+\frac{6}{5}y=g \)
\( \frac{8}{13}x+hy=k \)
In the given system of equations, g, h, and k are constants. The system has infinitely many solutions. What is the value of \( \frac{g}{k} \)?
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方程组与方程解的个数进阶
5x+3y=6
The given equation is one equation in a system of two linear equations. If the system of equations has at least one solution, which of the following equations could be the other equation in the system?
I. 7.5x+4.5y=9
II. 7.5x-4.5y=9
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方程组与方程解的个数进阶
4x - 6y = 10y +2
ty= \(\frac{1}{2}\)＋2x
In the given system of equations,t is a constant. If the system has no solution, what is the value of t?
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方程组与方程解的个数进阶
\( \frac{12x+16}{4}-\frac{s}{13}=r\left(x-9\right) \)
In the given equation, s and r are constants, and \( s>0. \) If the equation has infinitely many solutions, what is the value of s?
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方程组与方程解的个数进阶
\( -\frac{3}{19}rx+\frac{s}{8}=10-\frac{5}{57}x \)
In the given equation, r and s are constants. The equation has no solution. What is the value of r?
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方程组与方程解的个数进阶
3x+5y=89x+15y=24For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?
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方程组与方程解的个数基础
\( -12x+14y=36 \)
\( -6x+7y=-18 \)
How many solutions does the given system of equations have?
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