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
\( y=2x+10 \)
\( y=2x-1 \)
At how many points do the graphs of the given equations intersect in the xy-plane?
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
\( 3x=36y-45 \)
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
\( 66x=66x \)
How many solutions does the given equation have?
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
方程组与方程解的个数挑战
The equation \(9x+5=a(x+b)\), where \(a\) and \(b\) are constants, has no solutions. Which of the following must be true?
I. \(a=9\)
II. \(b=5\)
III. \(b\ne\frac{5}{9}\)
原库参考解析
方程组与方程解的个数挑战
\( 3\left(kx+13\right)=\frac{48}{17}x+36 \)
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
方程组与方程解的个数挑战
\( 2\left(kx-n\right)=-\frac{28}{15}x-\frac{36}{19} \)
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
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
-15x +25y =65
One of the two equations in a system of linear equations is given. The system has infinitely many solutions. Which of the following 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
3x +21 =3x +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
\( 4x+12=\frac{a(x+3)}{2} \)
In the given equation, a is a constant. If the equation has infinitely many solutions, what is the value of a?
原库参考解析
