How to plot the linear system solutions with multiple solutions?

In summary, the linear system of equations can be solved using Cramer's rule or Kronecker-Capelli's theorem. The solutions and equations can be plotted in cases where the system is consistent, resulting in either a point, a line, or a plane. The values of a and b determine the type of solution and the points on the graph.
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gruba
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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?
 
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  • #2
I didn't check the correctness of your result, but assuming all is right :

gruba said:
[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].

If it has a unique solution, so it's a point.

gruba said:
This gives [itex](x,y,z)=(1-y-z,y,z)[/itex]. How to plot the equations with intersection (line) in this case?

That means that the solution verifies ## x+ y + z - 1 = 0 ## when ##x,y,z## describe ##\mathbb{R}^3##.
It's a plane passing through (1,0,0), (0,1,0), and (0,0,1).

gruba said:
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?

We already discussed that in length yesterday :mad:
 
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What is the purpose of plotting a linear system?

The purpose of plotting a linear system is to visually represent the relationship between two or more variables in a linear equation. This allows for a better understanding of the behavior of the system and can help with making predictions and solving problems.

How do you plot a linear system?

To plot a linear system, you need to identify the variables and their corresponding equations, choose a suitable scale for the axes, and plot the points that satisfy each equation. Then, connect the points with a straight line to represent the relationship between the variables.

What is the significance of the slope in a linear system?

The slope in a linear system represents the rate of change between the two variables. It can indicate if the relationship is positive or negative and how steep the line is, which can provide valuable insights into the behavior of the system.

How can you determine the solution to a linear system from a graph?

The solution to a linear system can be determined from a graph by finding the point where the lines intersect. This point represents the values of the variables that satisfy all of the equations in the system and is known as the solution point.

What are some common applications of plotting a linear system?

Plotting a linear system has many real-world applications, such as in economics, engineering, and physics. It can be used to model and analyze relationships between variables, make predictions, and solve problems in various fields.

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