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
How many solutions does the equation \( 6x+4=6x+8 \)have?
原库参考解析
方程组与方程解的个数挑战
\( y=\frac{2}{5}x+7 \)
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进阶
\( 7x-2y=2y+5 \)
\( hy=4+7x \)
In the given system of equations, h is a constant. If the system has no solution, what is the value of h?
原库参考解析
方程组与方程解的个数挑战
How many solutions does the equation 11( x + 18) + 7x = 18( x +11) have?
原库参考解析
方程组与方程解的个数进阶
\( -3x+6rx=54 \)
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
\( 7x+5y=4 \)
\( 21x+15y=12 \)
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
How many solutions does the equation \( 7x-19=19-7x \) have?
原库参考解析
方程组与方程解的个数进阶
\( y=\frac{4}{7}x+5 \)
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
\( 8x+7y=3 \)
\( 40x+35y=15 \)
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
The graph of line g is shown in the xy-plane. Line k is defined by \( 165x+py=w \) , where p and w are constants. If line k is graphed in this xy-plane, resulting in the graph of a system of two linear equations, the system of two linear equations will have infinitely many solutions. What is the value of \( p+w \)?
原库参考解析
方程组与方程解的个数挑战
\( 7x+5y=2 \)
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.\( 10.5x+7.5y=3 \)
Ⅱ.\( 10.5x-7.5y=3 \)
原库参考解析
方程组与方程解的个数挑战
How many solutions does the equation \( 5x+3=5x \) have?
原库参考解析
