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节 2.1 线性方程组、消元法及几何直观
练习 练习
基础题.
1.
判断下列方程是否是关于变元\(x_1,x_2,x_3\)的线性方程。
-
-
\(\frac{1}{2}x_1-3x_2=4\)；
-
\(2^{x_1}+3^{x_2}+5^{x_3}=6\)；
-
\(2x_1-x_2^{-1}+3x_3=0\)；
-
\(\tan x_1 +\sin x_2+\cos x_3=3\)；
-
\(-x_1+2x_3=7x_1-2x_2-9\)。
2.
判断下列方程组是否是关于变元\(x,y\)的线性方程组。若是，求解该线性方程组。
-
\(\displaystyle \left\{\begin{array}{ccc}
2x^2-y^2&=&3,\\
-x^2+y^2&=&-1;
\end{array}\right.\)
-
\(\displaystyle \left\{\begin{array}{ccc}
3x-6y&=&3,\\
-x+2y&=&1;
\end{array}\right.\)
-
\(\displaystyle \left\{\begin{array}{ccc}
3x-6y&=&3,\\
-x+2y&=&-1.
\end{array}\right.\)
3.
求解下列线性方程组：
-
\(\displaystyle \left\{\begin{array}{rcl}
x_1-x_2+x_3&=&b_1,\\
x_2+2x_3&=&b_2,\\
x_3&=&b_3;
\end{array}\right.\)
-
\(\displaystyle \left\{\begin{array}{rcl}
x_1+2x_2-x_3+x_4&=&b_1,\\
x_3-2x_4&=&b_2,\\
x_4&=&b_3;
\end{array}\right.\)
-
\(\displaystyle \left\{\begin{array}{rcl}
x_1+x_2+x_3+x_4&=&1,\\
3x_1+2x_2+x_3+x_4&=&-3,\\
5x_1+4x_2+3x_3+3x_4&=&-1.
\end{array}\right.\)
