The systems of linear equations seen early in this lesson each had a single solution. In the Investigation, you saw that a single solution to a system of linear equations is not the only possibility. Systems of equations involving lines with the same slope are special cases.

  • Parallel lines have no points of intersection and the corresponding system of equations has no solution.

  • Coincident lines have an infinite number of points of intersection and the corresponding system of equations has an infinite number of solutions.


Coincident Lines
lines that lie on top of one another


For a pair of coincident lines, when a point is on one line, it must also be on the other line. This means each point on the lines is a point of intersection and is therefore a solution to the corresponding system of equations.

If lines have different slopes, they are guaranteed to cross in exactly one place unless the domain of either line is restricted.

Key Lesson Marker

 

Lines with different slopes have one point of intersection. This type of system has one solution. Lines with equal slopes and different y-intercepts are parallel and do not intersect. This type of system has no solution. Lines with equal slopes and equal y-intercepts are coincident lines and have an infinite number of points of intersection. This type of system has an infinite number of solutions.