The solution to a system of equations can be understood graphically by looking at where the graphs of the equations intersect on a coordinate plane.
Consider a system of two linear equations in two variables, \(x\) and \(y\):
\[ \begin{cases}
ax + by = c \\
dx + ey = f
\end{cases} \]
Each equation represents a straight line on the coordinate plane. The solution to the system is the point where these two lines intersect.
There are three possible scenarios:
1. The lines intersect at a single point. This point represents the unique solution to the system of equations.
2. The lines are parallel and do not intersect. In this case, there is no solution to the system because the equations represent two lines that never meet.
3. The lines coincide, meaning they are essentially the same line. In this case, there are infinitely many solutions to the system because any point on the line is a solution.
Graphically, you can visualize these scenarios by plotting the lines corresponding to each equation and observing their relationship on the coordinate plane.
