(a) 证明方程组

没有唯一解。

(b) 证明 (a) 中的方程组有解。从几何角度解释这种情况。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

检查三行系数之间的关系。第三行可由前两行线性组合得到，所以系数行列式为零。

For the coefficient matrix, $R_3=2R_2-R_1$. For example, expanding its determinant gives

The third row is therefore dependent on the first two. The rows are therefore linearly dependent, so the coefficient determinant is zero. Hence the system does not have a unique solution.

同样的行关系在右端也成立，因此第三式不增加条件。用前两式消元，找出一条由自由参数表示的公共直线。

The third equation is twice the second minus the first, including its right-hand side $2\cdot1-1=1$. Subtracting the first equation from the second gives $3x+3y+3z=0$, hence $x+y+z=0$. Combining this with the first gives $y+2z=1$. Thus the solution set is

The system is consistent and the three planes share this common line.