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
\( \frac{-20x-47+m}{2}=k(x+7) \)
In the given equation, k and m are constants. 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进阶
How many solutions does the equation \( 35x−25=5(7x−5) \) have?
原库参考解析
方程组与方程解的个数挑战
\( y=4x+36 \)
One of the two equations in a system of linear equations is given. The system has infinitely many solutions. 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
\( x-4y=-16 \)
\( 4x-16y=0 \) \( 3x-12y=48 \)
The given system of equations is graphed in the xy-plane. How many solutions\( (x,\ y) \) does the system of equations have?
原库参考解析
方程组与方程解的个数挑战
\( \frac{8cx+7}{5}=4x-7 \)
In the given equation, c is a constant. If the equation 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
\( y=5x+18 \)
\( y=5x-18 \)
How many solutions does the given system of equations have?
原库参考解析
方程组与方程解的个数进阶
\( 5x+8y=30 \)
\( rx+24y=90 \)
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
\( -3x+24px=96 \)
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
方程组与方程解的个数挑战
How many solutions does the equation \( 10(x+15)+5x=15(x+10) \) have?
参考答案
方程组与方程解的个数挑战
\( a(9−x)=36−4x \)
In the given equation, a is a constant. The equation has exactly one solution. Which of the following CANNOT be the value of a?
参考答案
方程组与方程解的个数挑战
\( 2x+5y=7 \)
\( 8x+20y=28 \)
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?
参考答案
方程组与方程解的个数挑战
How many solutions does the equation \( 7x+3x+12=10x+5+7 \) have?
参考答案
