Question

How do you know by looking at a graph that a system of linear equations has infinite solutions?

Answer 1

Hint: For linear systems of equations we have three cases $ (1) $ No solutions $ (2) $ Finite solutions $ (3) $ Infinite solutions and we have a condition which has to be satisfied for each category of solution. For a system of linear equations to have an infinite number of solutions they have to intersect at an infinite number of points. From this point you definitely got the idea.

Complete step-by-step answer:
To tell just by looking at a graph that a system of linear equations has infinite solutions. First we have to be clear about the type of solutions to a system of linear equations.

 $ (1) $ No solutions: A system of linear equations has no solutions if the lines formed by those linear equations are parallel in each other. It means that the system of linear equations has no solution. Here is an example:

In the above graph both the lines are running parallel to each other infinitely. So they have no solution.

 $ (2) $ Finite solutions: A system of linear equations has finitely many solutions if the lines formed by those linear equations are not parallel. It means that the system of linear equations has finite solutions. Here is an example:

The point of intersection of both lines is the solution of the system of equations.

 $ (3) $ Infinite solutions: A system of linear equations has infinitely many solutions if the lines formed by those linear equations are coincident on each other. It means that the system of linear equations has infinite solutions. Here is an example:

You can see two lines overlapping in this graph one is of green colour and the other is of red colour

Note: A mistake student does while telling a system of linear equations just by looking at graph has infinite solutions even they see two lines coincident of system of linear equations, but for a system of linear equations to have infinite solutions then all lines formed from system of linear equations has to be coincident.