- #1
gruba
- 206
- 1
Homework Statement
Solve the linear system of equations:
[itex]ax+by+z=1[/itex]
[itex]x+aby+z=b[/itex]
[itex]x+by+az=1[/itex]
for [itex]a,b\in\mathbb R[/itex]
and plot equations and solutions in cases where the system is consistent.
Homework Equations
-Cramer's rule
-Kronecker-Capelli's theorem
The Attempt at a Solution
Using Cramer's rule, we find the determinant of the system and determinant for each variable:
[itex]D=b(a-1)^2(a+2)[/itex]
[itex]D_x=b(a-b)(a-1)[/itex]
[itex]D_y=(a-1)(ab+b-2)[/itex]
[itex]D_z=b(a-1)(a-b)[/itex]
[itex][/itex]
For [itex]b\neq 0 \land a\neq 1\land a\neq -2\Rightarrow D\neq 0[/itex] system has unique solution:
[itex](x,y,z)=\left(\frac{a-b}{(a-1)(a+2)},\frac{ab+b-2}{b(a-1)},\frac{a-b}{(a-1)(a+2)}\right)[/itex].
How to plot the equations with intersection (point) in this case?
Second case, [itex]a=1[/itex].
Solvind the system using Kronecker-Capelli's theorem gives:
[itex]b=1\Rightarrow[/itex] infinitely many solutions.
[itex]b\neq 1\Rightarrow[/itex] the system is inconsistent.
This gives [itex](x,y,z)=(1-y-z,y,z)[/itex].
How to plot the equations with intersection (line) in this case?
Third case, [itex]a=b=-2\Rightarrow[/itex] infinitely many solutions.
[itex](x,y,z)=\left(z,\frac{-z-1}{2},z\right)[/itex].
How to plot the equations with intersection (line) in this case?