When you solve a system of equations graphically, you find an image of the solution by drawing the equations in a coordinate system and figuring out where they intersect. It can be useful to imagine systems of equations as graphs coming together. Then you always have a clear idea of what is going on.

The equations will then be straight lines in the form $y=ax+b$.

When the System of Equations Has No Solution

When the graphs have the same slope $a$, they are parallel. As long as the constant term $b$ in the two functions is different, these lines will never intersect. In short: No intersection = no solution.

When the System of Equations Has One Solution

When the lines have different slopes $a$, they will intersect at one point. This point is called the intersection. In short: One intersection = one solution. The intersection is made up of a value from the first axis (often $x$) and one value from from the second axis (often $y$), so you can find the solution by reading off the coordinates where the two lines intersect. When you solve systems of equations through algebra, these are the exact coordinates you will find.

When the System of Equations Has an Infinite Number of Solutions

When the two graphs have the same slope $a$, they’re parallel. When the constant term $b$ is also the same in both functions, these lines will be identical, and on top of each other. They intersect at all points on the graph, which is an infinite number of points. In other words, an infinite number of intersections = an infinite number of solutions. The solution is therefore all the points on the line, and we write it as $S=ℝ$.

It’s important to notice that the answer is made up of both the $x$ value and $y$ value together. For that reason, it’s smart to think about the solution as the coordinates at the intersection of the two graphs.