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
\( -3x+6rx=42 \)
In the given equation, r is a constant. The equation has no solution. What is the value of r?
原库参考解析
方程组与方程解的个数进阶
\( 7x+21=7x+k \)
In the given equation, k is a constant. The equation has infinitely many solutions. What is the value of k ?
原库参考解析
方程组与方程解的个数挑战
\( 3x=45y-27 \)
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挑战
How many solutions does the equation \( 2x-3=2x \) have?
原库参考解析
方程组与方程解的个数挑战
\( x\left(r-7\right)+3=17x+25 \)
In the given equation, r is a positive integer. If the given equation has exactly one solution, what CANNOT be the value of r?
原库参考解析
方程组与方程解的个数挑战
\( y=\frac{2}{7}x+3 \)
One of the two equations in a system of linear equations is given. The system has infinitely many solutions. If the second equation in the system is \( y=mx+b \), where m and b are constants, what is the value of b?
原库参考解析
方程组与方程解的个数挑战
\( x\left(r-7\right)+3=19x+25 \)
In the given equation, r is a positive integer. If the given equation has exactly one solution, what CANNOT be the value of r?
原库参考解析
方程组与方程解的个数进阶
\( 5\left(kx-n\right)=-\frac{65}{14}x-\frac{85}{18} \)
In the given equation, k and n are constants and \( n>1 \) . The equation has no solution. What is the value of k?
原库参考解析
方程组与方程解的个数挑战
\( 16x-20y=6y+8 \)
\( ty=8x+\frac{1}{5} \)
In the given system of equations, t is a constant. If the system has no solution, what is the value of t?
原库参考解析
方程组与方程解的个数进阶
\( 3x+9y=10 \)
\( rx+18y=20 \)
In the given system of equations, r is a constant. The system has infinitely many solutions. What is the value of r?
原库参考解析
方程组与方程解的个数挑战
\( 5x+3=rx+7 \)
In the given equation, r is a constant. The equation has no solution. What is the value of r?
原库参考解析
方程组与方程解的个数挑战
\( \frac{5}{9}(x-k)=k\left(x-\frac{9}{5}\right) \)
In the given equation, \( k \) is a constant. The equation has no solution. What is the value of \( k \)?
原库参考解析
