When a system of linear equations has two equations and two variables, the system can have zero, one or infinitely many solutions.
Recall that the solution to a system is the point where the lines intersect. If a system has no solution, it must be then that the lines never intersect. In fact, the lines must be parallel, meaning that they have the same slope and different -intercepts. An example of one such system is
In contrast to parallel lines, lines that intersect once must have unequal slopes. For example, the system shown must have exactly one solutin as the two lines have different slopes.
Such lines are said to be coincidental, and, as they have the same slope and -intercept, they are different versions of the same line. One example of a system that has an infinite number of solutions is
Find the value of that ensures the system has infinitely many solutions.
A system of equations has infinitely many solutions when the equations describe the same line. In other words, the lines must have the same slope and -intercept. The first line has a slope of Thus, the slope of the second line must also be This means, The system can now be written Notice that the -intercepts of the lines have already been given. They are and Since these values are different, and they cannot be changed, the lines are not coincidental. Therefore, the system will never have infinitely many solutions. In fact, since the lines are parallel, the system has no solution.
Determine the number of solutions the system has.
Determine if the point is a solution to the system
Add and subtract terms
Determine the number of solutions the system has algebraically.
To algebraically find the number of solutions to the system, either the substitution method or the elimination method can be used. Since there are equally many and terms in both equations, the elimination method is most convenient. Therefore, the second equation should be subtracted from the first. This resulted in a contradiction, so the system has no solution.