4. Systems of Equations
Systems of Linear Equations
Ex 1.
Solve the system of equations:
\[
\begin{aligned} 2x+3y&=5,\\ x-4y&=-2. \end{aligned}
\]
using the methods: Cramer's rule, Gaussian elimination, and inverse matrix.
Ex 2.
Solve the system of three equations with three unknowns:
\[
\begin{aligned} x+y+z&=6,\\ 2x-y+3z&=14,\\ -x+2y-z&=-2. \end{aligned}
\]
using the methods: Cramer's rule, Gaussian elimination, and inverse matrix.
Ex 3.
\(\star\) Consider the parametric system dependent on \(\lambda\):
\[
\begin{aligned} x+\lambda y&=1,\\ 2x+(1+\lambda)y&=3. \end{aligned}
\]
Determine the values of \(\lambda\) for which the system has one solution, infinitely many solutions, or no solution.
Ex 4.
For the coefficient matrix
\[
A=\begin{pmatrix}1 & 1 & 1\\ 0 & 2 & -1\\ 2 & -1 & 3\end{pmatrix}
\]
and the right-hand side vector \(b=(4,1,3)^{\top}\), solve \(Ax=b\) and check the result by substitution.
Ex 5.
Solve the systems of equations:
a)
\[
\begin{cases}
x_1 + 4x_2 + 3x_3 = 1, \\
2x_1 + 5x_2 + 4x_3 = 4, \\
x_1 - 3x_2 - 2x_3 = 5;
\end{cases}
\]
b)
\[
\begin{cases}
x_1 - 2x_2 - 3x_3 = 2, \\
x_1 - 4x_2 - 13x_3 = 14, \\
-3x_1 + 5x_2 + 4x_3 = 0;
\end{cases}
\]
c)
\[
\begin{cases}
x_1 - 2x_2 - 3x_3 = 2, \\
x_1 - 4x_2 - 13x_3 = 14, \\
-3x_1 + 5x_2 + 4x_3 = 2;
\end{cases}
\]
d)
\[
\begin{cases}
-4x_1 + 3x_2 + 2x_3 = -2, \\
5x_1 - 4x_2 + x_3 = 3;
\end{cases}
\]
e)
\[
\begin{cases}
-4x_1 + 3x_2 = 2, \\
5x_1 - 4x_2 = 0, \\
2x_1 - x_2 = a;
\end{cases}
\]
f)
\[
\begin{cases}
4x_1 + 5x_3 = 6, \\
x_2 - 6x_3 = -2, \\
3x_1 + 4x_3 = 3;
\end{cases}
\]
g)
\[
\begin{cases}
3x_1 - x_2 - 2x_3 = 2, \\
2x_2 - x_3 = -1, \\
3x_1 - 5x_2 = 3;
\end{cases}
\]
h)
\[
\begin{cases}
-x_1 + 2x_2 + 3x_3 = 0, \\
x_1 - 4x_2 - 13x_3 = 0, \\
-3x_1 + 5x_2 + 4x_3 = 0;
\end{cases}
\]
i)
\[
\begin{cases}
x_1 + x_2 + x_3 = 0, \\
2x_1 + 4x_2 + 3x_3 = 0, \\
4x_2 + 4x_3 = 0;
\end{cases}
\]
j)
\[
\begin{cases}
x_1 + x_2 + x_3 = -2, \\
2x_1 + 4x_2 - 3x_3 = 3, \\
4x_2 + 2x_3 = 2;
\end{cases}
\]
k)
\[
\begin{cases}
4x_1 + 4x_3 = 8, \\
x_2 - 6x_3 = -3, \\
3x_1 + x_2 - 3x_3 = 3;
\end{cases}
\]
l)
\[
\begin{cases}
5x_1 - 3x_2 = -7, \\
-2x_1 + 9x_2 = 4, \\
2x_1 + 4x_2 = -2;
\end{cases}
\]