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挑战
\( y=\frac{1}{2}x+8 \)
y = cx +10
In the system of equations above, c is a constant. If the system has no solution, what is the value of c ?
原库参考解析
方程组与方程解的个数挑战
y =2x +1
y = ax -8
In the system of equations above, a is a constant. If the system of equations has no solution, what is the value of a ?
原库参考解析
方程组与方程解的个数挑战
8x + 7y =9
24x + 21y =27
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
One of the two equations in a linear system is 2x +6y =10. The system has no solution. Which of the following could be the other equation in the system?
原库参考解析
方程组与方程解的个数挑战
2x +3y =7
10x +15y =35
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
y = 6x + 18
One 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?
原库参考解析
方程组与方程解的个数挑战
If a new graph of three linear equations is created using the system of equations shown and the equation x + 4y = -16, how many solutions (x , y ) will the resulting system of three equations have?
原库参考解析
方程组与方程解的个数挑战
\( \frac{3}{2}y-\frac{1}{4}x=\frac{2}{3}-\frac{3}{2}y \)
\( \frac{1}{2}x+\frac{3}{2}=py+\frac{9}{2} \)
In the given system of equations, p is a constant. If the system has no solution, what is the value of p ?
原库参考解析
方程组与方程解的个数挑战
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 ?
原库参考解析
方程组与方程解的个数挑战
In the system of equations below, a and c are constants.
\( \frac{1}{2}x+\frac{1}{3}y=\frac{1}{6} \)
ax + y = c
If the system of equations has an infinite number of solutions ( x , y ) , what is the value of a ?
原库参考解析
方程组与方程解的个数挑战
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 \( r \)?
原库参考解析
方程组与方程解的个数挑战
−3x + 21px = 84
In the given equation, p is a constant. The equation has no solution. What is the value of p ?
原库参考解析
