Inconsistent Equation: Definition & Examples

Instructor: Laura Pennington

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

Inconsistent equations are a specific type of system of equations. In this lesson, we will learn about inconsistent equations and how to identify them. When you've finished the lesson, you will be able to test your knowledge with a quiz.

System of Equations

Before we can get to inconsistent equations, we first need a quick review of systems of equations and how to solve them. A system of equations is a collection of two or more equations involving the same variables. These systems come in handy when you are facing a problem where there is more than one unknown quantity.

For example, suppose we are looking for two numbers such that five times the first number added to two gives the second number, and if you subtract two times the second number from 10 times the first number, you get 12. In this example, we have two unknowns. Let x = the first number and y = the second number. We are given that five times the first number added to two gives the second number, therefore 5x + 2 = y. We are also given that if you subtract two times the second number from 10 times the first number, you get 12, therefore 10x - 2y = 12. This gives the following system of equations.

5x + 2 = y

10x - 2y = 12

Recall that there are two ways to solve a system of equations: substitution or elimination. These methods are summarized here:

Inconsistent equations 1

Inconsistent equations 2

We can also observe the graphs of each of the equations in a system. The solutions to the system are where the graphs intersect.

Inconsistent Equations

When it comes to systems of equations, the system either has a solution or it doesn't. In this lesson, we are going to concentrate on the latter possibility. When a system has no solution, it is called inconsistent. It just so turns out that our initial example is an example of an inconsistent system of equations. There are no two numbers that satisfy the description given. In other words, no two numbers exist such that five times the first number added to two gives the second number, and if you subtract two times the second number from 10 times the first number, you get 12.

Determining if a System is Inconsistent

To determine if a system of equations is inconsistent, you would go about solving it as you would any system of equations. If the system is inconsistent, then at some point, you will run into a statement that doesn't make sense, such as 0 = 3. If this happens, you have inconsistent equations. Consider our example. Suppose we try to solve the system using substitution. We have that y = 5x + 2, so we plug 5x + 2 in for y into the second equation, then try to solve for x.

10x - 2(5x + 2) = 12 distribute the -2

10x - 10x - 4 = 12 add 4 to both sides

10x - 10x = 16 simplify the left hand side of the equation

0 = 16

Zero can't equal sixteen, so the statement 0 = 16 makes no sense. Therefore, the system is inconsistent and has no solution.

We can also observe that the system is inconsistent by analyzing the graphs of each of the equations in the system.

Graph of an Inconsistent System
Inconsistent equation 3

Notice that the two lines are parallel. This makes sense, because the solution to a system of equations is where their graphs intersect. Two parallel lines never intersect, so the system has no solution and is inconsistent.

Example

Suppose you go to the store, and you buy 2 pounds of apples and 1 pound of strawberries. Your total bill is $3.00. A friend of yours goes with you, and she buys 6 pounds of apples and 3 pounds of strawberries. Her total bill is $15.00. Explain why it is impossible that you and your friend paid the same amount per pound of apples and per pound of strawberries.

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