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
\( \frac{1}{9}x-\frac{3r}{7}y=34 \)
One of the equations in a system of two linear equations is given, where r is a nonzero constant. The system has no solution. If the other equation in the system is graphed in the xy-plane, what is the slope of the graph?
原库参考解析
方程组与方程解的个数挑战
How many solutions does the equation\( 7\left(x+12\right)+5x=12\left(x+7\right) \) have?
原库参考解析
方程组与方程解的个数进阶
\( 45x+33y-200=-359 \)
\( 2wx+11y=-53 \)
In the given system of equations, w is a constant. The system has infinitely many solutions. What is the value of w?
原库参考解析
方程组与方程解的个数进阶
How many solutions does the equation \( 17\left(x-16\right)=-16\left(x+17\right) \) have?
原库参考解析
方程组与方程解的个数进阶
\( y=\frac{2}{5}x+2 \)
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
\( a\left(7x-14\right)+5a=7\left(ax-2a\right)-35 \)
In the given equation, a is a constant. The equation has infinitely many solutions. What are all possible values of a?
原库参考解析
方程组与方程解的个数进阶
The graph of line h is shown in the xy-plane. Line k (not shown) is defined by \( sx+40y=t, \) where s and t are constants. If line k is graphed in this xy-plane, the result is a graph of two linear equations. This system of two linear equations has no solution. Which of the following is NOT a possible value of t?
原库参考解析
方程组与方程解的个数挑战
\( 13\left(x-n\right)=13y+13n \)
One of the equations in a system of two linear equations is given, where n is a positive constant. 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进阶
y= -4x-2
One of the two equations in a system of linear equations is given. The system has no solution. Which of the following could be the second equation in this system?
原库参考解析
方程组与方程解的个数进阶
6x +28 =6x +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
-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挑战
The graph of line g is shown in the xy-plane. Line k is defined by 195x + 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
