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
\( 18(x+5)=2(x+c)+16x \)
In the given equation, c is a constant. The equation has infinitely many solutions. What is the value of c?
In the given equation, c is a constant. The equation has infinitely many solutions. What is the value of c?
参考答案
方程组与方程解的个数挑战
\( 4x+9y=30 \)
\( rx+18y=60 \)
In the given system of equations, r is a constant. The system has infinitely many solutions. What is the value of r?
\( rx+18y=60 \)
In the given system of equations, r is a constant. The system has infinitely many solutions. What is the value of r?
参考答案
方程组与方程解的个数进阶
The graph of a system of three equations is shown.
The system consists of two linear equations and one quadratic equation. How many solutions does this system have?
The system consists of two linear equations and one quadratic equation. How many solutions does this system have?
参考答案
方程组与方程解的个数挑战
\( 8x+7y=5 \)
The given equation is one equation in a system of two linear equations. If the system of equations has at least one solution, which of the following equations could be the other equation in the system?
I. \( 12x+10.5y=7.5 \)
II. \( 12x-10.5y=7.5 \)
The given equation is one equation in a system of two linear equations. If the system of equations has at least one solution, which of the following equations could be the other equation in the system?
I. \( 12x+10.5y=7.5 \)
II. \( 12x-10.5y=7.5 \)
参考答案
方程组与方程解的个数挑战
Which system of linear equations has no solution?
参考答案
方程组与方程解的个数基础
\( y=3x+7 \)
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?
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?
参考答案
方程组与方程解的个数进阶
Which equation has no solution?
参考答案
方程组与方程解的个数进阶
If a new graph of three linear equations is created using the system of equations shown and the equation \( x+4y=-32 \), how many solutions (x, y) will the resulting system of three equations have?
参考答案
方程组与方程解的个数挑战
\( c(x-2)=-5(x+k) \)
In the given equation, c and k are constants. The equation has exactly one solution. Which of the following statements must be true?
In the given equation, c and k are constants. The equation has exactly one solution. Which of the following statements must be true?
参考答案
方程组与方程解的个数基础
\( -5x+10y=75 \)
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?
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?
参考答案
方程组与方程解的个数挑战
\( \frac{5}{4}(x-k)=k\left(x-\frac{4}{5}\right) \)
In the given equation, k is a constant. The equation has no solution. What is the value of k?
In the given equation, k is a constant. The equation has no solution. What is the value of k?
参考答案
方程组与方程解的个数进阶
At how many points do the graphs of the equations \( y=x+13 \) and \( y=4x \) intersect in the xy-plane?
原库参考解析
