A linear system of equations will have a single solution so long as the equations represent nonparallel lines. For example, consider the system 8x + 4y = 12 and -5x + 5y = 6. Using algebra, you can write these two equations in the equivalent forms of y = -2x + 3 and y = x + 1.2. These are the equations of two lines with different slopes; therefore, they intersect at a single point. The solution for this particular example is x = 0.6 and y = 1.8 --- equivalent to the point (0.6, 1.8) in coordinate form.
Linear systems that represent parallel lines have no solution; two parallel lines will never cross each other and thus have no intersection point. A set of parallel lines will have the same slope but different y intercepts. For instance, consider the system -2x + y = -3 and 4x - 2y = 10. Using algebra, you can rewrite these equations as y = 2x - 3 and y = 2x - 5. The first line has a slope of 2 and a y-intercept of -3; the second has a slope of 2 and a y-intercept of -5. Since these lines are parallel, the system has no solution.
When a system of linear equations consists of the same equation repeated twice, the system has infinitely many solutions because a line has infinitely many points in common with itself. Consider the linear system -9.1x + 2.8y = 7 and 63.7x - 19.6y = -49. At first, these may appear distinct equations; however, after you simplify them, you obtain y = 3.25x + 2.5 for both equations. Since they represent the same line, this set of equations has infinitely many solutions. For example, the points (0, 2.5), (2, 9) and (10, 35) are just three solutions to the system, though you can find infinitely many more.
For systems of linear equations with more than two variables, the number of solutions is still either zero, one or infinitely many --- a property of linear equations. Only in nonlinear systems can you have two, three or four solutions. Linear systems in three variables represent planes in three-dimensional space.