Short Answer 
To identify a system of equations with one solution, the graphs of the equations must intersect at a single point, indicating a unique pair of x and y values. Examples demonstrate this concept by showcasing pairs of equations that meet at distinct points, and visual representation through graphing can further validate the presence of a single solution.
Step 1: Understanding Intersections
To identify a system of equations with one solution, it is essential to understand that the graph will display two lines intersecting at a single point. This point represents the exact x and y values shared by both equations, illustrating that there is only one solution. In graphical terms, the lines should not be parallel or coincident, as these scenarios would indicate either no solution or infinitely many solutions.
Step 2: Analyzing Examples
Let’s explore two examples to clarify how the intersection point signifies a single solution. Consider the following systems of equations:
- Equation 1: y = 2x + 1
- Equation 2: y = -3x + 4
When graphed, these lines intersect at a point, proving they have a single solution together. Similarly, in this second example:
- Equation 1: y = -x + 3
- Equation 2: y = x – 1
The lines again cross at only one point, confirming that this set of equations also has one solution.
Step 3: Visual Representation
To visually confirm the presence of a single solution, you can plot the equations on a graph. Use graph paper or digital tools to draw the lines according to their equations. Observe where the lines meet, ensuring that:
- The lines should intersect at one distinct point.
- This intersection should not correspond to multiple points or parallel lines.
By verifying the graphical representation, it becomes clearer that the system of equations in question possesses exactly one solution.