Question

Describe the graph of a linear system that has the given number of solutions. Sketch an example. infinitely many solutions

Short answer

A linear system has infinitely many solutions when all the lines represented by the equations in the system coincide completely. A sketch for such system would show a single line, as all equations represent the same line and hence all points on that line are solutions.

Step-by-step solution

Understanding the System

A system is said to have infinitely many solutions when all the equations represent the same line. In this case, there would be an infinite number of points of intersection since all points on the line are solutions to the system.

Sketch of the System

An example of such a system would be : \(y = 2x + 1\) and \(2y = 4x + 2\). As we can see the second equation is just a multiple of the first one, therefore they are essentially the same line. When we draw these two equations, it shows up as only one line on the graph. Therefore, since the lines coincide, any point on that line are solutions - hence there are infinitely many.