Solving Special Systems of Linear Equations

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Systems of linear equations with no solution or infinitely many solutions are special systems of linear equations. This lesson will explore what happens when we try to solve these special systems of linear equations both graphically and algebraically.

Special Systems of Linear Equations

Suppose you are sitting in algebra class and a funny thing happens - your teacher catches your attention! This happens when she says that she will cut class short by 5 minutes if anyone in the class can find two numbers, x and y, such that both of the following equations are true:

y = 3x + 2

y - 3x = 0

You get excited, because you recall that this is a system of linear equations, and you know how to solve these!


A system of linear equations is a set of linear equations with the same variables. The solution to a system of linear equations consists of the values of the variables that make all of the equations in the system true.

To find the solution of a system of linear equations algebraically, we can use substitution or elimination. For simplicity, we will just use substitution in this lesson, which involves the following steps:

  1. Solve both equations for the same variable, say y.
  2. Set the two expressions for that variable equal to one another.
  3. Solve the resulting equation for the other variable, call it x.
  4. Plug the value you find for x into either of the original equations and solve for y.

Let's solve this system of equations, so you can go home early! First, we solve both equations for y. This only needs to be done for the second equation.

y - 3x = 0

If we add 3x to both sides to solve for y, we get:

y = 3x

Okay, we have y = 3x + 2 and y = 3x, so we set 3x + 2 equal to 3x and solve for x.

3x + 2 = 3x Subtract 3x from both sides.
3x + 2 - 3x = 3x - 3x Simplify
2 = 0 What the heck?

Huh? That doesn't make any sense, because 2 can't be equal to 0! As it turns out, your teacher pulled a fast one on you. You see, this system is a special type of system that has no solution. Ugh! So much for going home early!

There are two types of special systems of linear equations; those with no solution, and those with infinitely many solutions. Let's consider what happens when we try to solve these types of systems both graphically and algebraically.

Solving Graphically

Graphically speaking, the solution to a linear system of equations is the point at which the graphs of the equations in the system intersect. Let's think about this. If a system has no solution, what do you think that means for the graph of the system? You guessed it! Since the system has no solution, the graphs of the equations in the system don't intersect.


See how the graphs of the equations in the system your teacher gave you are parallel, so they don't intersect? This tells us there is no solution to the system.

Now, consider a system with infinitely many solutions. When this is the case, the equations in the system are the same equation, just in different forms, so all of the solutions to one equation are a solution to the other as well. What do we think this means graphically? If you are thinking the graphs of the equations in the system would be the same graph, you are correct! Since they are the same equation, they have the same graph.


We see that systems with infinitely many solutions have infinitely many intersection points.

Solving Algebraically

Let's consider solving algebraically now. Notice that while solving the system with no solution that your teacher gave you, you ran into a statement that made no sense (2 = 0). This is what will always happen when attempting to solve a system with no solution algebraically. You will end up with a statement that is false, such as 5 = 1 or 0 = 4. When this happens, you can stop the solving process and state that the system has no solution.

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