Question

Describe the graph of a linear system that has the given number of solutions. Sketch an example. exactly one solution

Short answer

The graph of a linear system that has exactly one solution consists of two intersecting lines. These lines cross at a unique point, which is the solution to the system. An example can be the equations \(y = 2x + 3\) and \(y = 3x + 1\), where the lines intersect at the point \((-2/2, 1)\).

Step-by-step solution

Describing the Graph

In a system of linear equations, if it's stated to possess exactly one solution, it implies that the graphs of the linear equations would intersect at exactly one point. This happens when the lines representing the equations are not parallel, and they cross each other at one particular point. This point of intersection is the solution to the system.

Sketching the Graph

Consider a set of two linear equations, such as \(y = 2x + 3\) and \(y = 3x + 1\). In the graph, you start by plotting both equations on the Cartesian coordinate system. The first equation produces a straight line and so does the second. Because these two lines are not parallel (they don't have the same slope), they intersect at a unique point. This point is the system's only solution.