Answer to Question #305278 in Linear Algebra for Nhlonipho Manganyi

Question

9.) Look up what is meant by a system of linear equations.

A known fact of solutions of systems of linear equations is that only one the following options can hold :

(i) No solution possible

(ii) A unique solution can be found

(iii) The system has infinite solutions.

Consider that two straight lines form a linear system.

Interpret what happens geometrically to the straight lines to get each case of the solution types given

abo

Answer 1

Solution (i)

If we have two straight lines such that there is no solution. It means that the two lines never intersect each other. There could be two reasons:

(1) The lines are in 2D and parallel, then for sure, they will never intersect each other.

For example, two parallel lines are

$2x+y=4$

$2x+y=-2$

The plot below shows that they never intersect each other.

The plot can be found at https://www.desmos.com/calculator/x9ep35tmeg

(2) The lines are in 3D and not parallel, then they will cross each other over another. Just like one road on the ground and the second road passing over the first one but over a bridge.

The diagram below shows two lines $L_{1}$ and $L_{2}$, which are not parallel and cross each other without intersecting.

Solution (ii)

If there is a unique solution, it means that the two lines are not parallel. Then the point where both intersect is called the unique solution.

The diagram below shows two straight lines

$x+y=-4$

$x-2y=5$

They are not parallel and have a unique solution $(-1,-3)$, the point of intersection between them.

https://www.desmos.com/calculator/os4qwsb6t6

The same holds in 3D.

Solution (iii)

In this case, the two lines are over one another.

consider the two lines

$x-2y=4$

$2x-4y=8$

Part of these two lines is plotted as shown in the figure.

We can see that every point on 1st line is also on the second line. Therefore there is an infinite number of solutions between the two lines.

We say that the two lines are the same.

https://www.desmos.com/calculator/wy3nwbb6se