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进阶
\( 7x+6y=5 \)
\( 28x+24y=20 \)
For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?
知序中文精讲
方程组与方程解的个数进阶
31( x - n) = 31y +31n
One of the equations in a system of two linear equations is given, where n is a positive constant.The system has no solution. Which equation could be the second equation in this system?
知序中文精讲
方程组与方程解的个数挑战
\[\frac{7}{8}y-\frac{5}{8}x=\frac{4}{7}-\frac{7}{8}y\]\[\frac{5}{4}x+\frac{7}{4}=py+\frac{15}{4}\]In the given system of equations, \(p\) is a constant. If the system has no solution, what is the value of \(p\)?
知序中文精讲
方程组与方程解的个数挑战
\( \frac{5}{8}x+\frac{7}{8}y=\frac{4}{7}-\frac{7}{8}y \)
\( \frac{5}{4}x+\frac{7}{4}y=py+\frac{5}{2} \)
In the given system of equations, p is a constant. If the system has no solution, what is the value of p?
知序中文精讲
方程组与方程解的个数进阶
48x - 72y = 30y + 24
ry= \(\frac{1}{6}\)- 16x
In the given system of equations,r is a constant.If the system has no solution, what is the value of r?
知序中文精讲
方程组与方程解的个数基础
\( 8x-3y=3y+9 \)
\( hy=2+8x \)
In the given system of equations, h is a constant. If the system has no solution, what is the value of h?
知序中文精讲
方程组与方程解的个数基础
How many solutions does the equation 7x-8=7x have?
原库参考解析
方程组与方程解的个数挑战
5x+4y=3
15x+12y=9
For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?
原库参考解析
方程组与方程解的个数挑战
5x=75y-125
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?
原库参考解析
方程组与方程解的个数进阶
For a linear relationship between x and y , the table shows 4 values of x and their corresponding values of y.
The graph of a system of two linear equations is shown. If the linear relationship represented by the table is graphed in this xy-plane, resulting in the graph of a system of three linear equations, how many solutions (x, y) will the system of three linear equations have?
原库参考解析
方程组与方程解的个数进阶
How many solutions does the equation 8x +5=8x＋ 2 have?
原库参考解析
方程组与方程解的个数挑战
y=5xy=2x+2How many solutions does the given system of equations have?
原库参考解析
