When we graph a system of equations, the solution is the point of intersection. What if the lines never intersect? What if the two reduce to be the same exact equation? Let's look at these situations individually.

No solution

Here we have two parallel lines that will never intersect.

Since there will never be an intersection, the system of equations

{y = 2 x + 1 y = 2 x + 4

has no solution.

Infinitely many solutions

Let's look at the following system of equations:

{y = 3 x - 1 2 y = 6 x - 2

and try to solve by graphing. Before we can graph the system, we need to make sure that both equations are in a slope-intercept form. The first equation already is, but the second equation needs to be divided by

2 .

{y = 3 x - 1 2 y = 6 x - 2 (I) (II)

DivEqn

$(II):$ $LHS / 2 = RHS / 2$

{y = 3 x - 1 y = 3 x - 1

Now we are ready to graph the equations.

Where do the lines intersect? How could you choose only one point of intersection? The lines are exactly the same and therefore intersection at an infinite number of points. When this happens, we say that there are infinitely many solutions.

Exercise