   SEARCH HOME Math Central Quandaries & Queries  Question from Ryan, a student: When is it possible to get no solution when solving a linear system? Ryan,

When solving linear systems with two variables you are often given 2 equations. Each of the equations represents a line in 2-space. For example

4 = 2x - y
2 = -3x + 7y

The solution is where the two lines intersect. So you have no solution precisely if there is no intersection. In what circumstances would you have a situation where two lines in 2-space don't intersect?

If you have three equations in two variables this represents three lines in 2-space. The solution is where the three lines intersect. So you have no solution precisely if there is no intersection, that is no point where all three lines meet. In what circumstances would you have a situation where three lines in 2-space don't intersect at a point?

What about n lines in 2-space?

Hope this helps.
Sue, Penny and Melanie

Hi Ryan,
From a graphical standpoint, the solution to a linear system of equations is the point where the lines intersect. Therefore, from a graphical standpoint, "no solution" would correspond to a system of parallel lines, or lines with the same slope but different y-intercepts.

If solving the system by elimination, a system with "no solution" would be one in which eliminating one variable from the system eliminates both variables and results in a statement of equality that is false, such as 2=0.

Hope this helps,
Leeanne     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.