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High School Algebra I: Tutoring Solution25 chapters | 257 lessons

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.

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 5*x* + 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 10*x* - 2*y* = 12. This gives the following system of equations.

5*x* + 2 = *y*

10*x* - 2*y* = 12

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

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

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.

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* = 5*x* + 2, so we plug 5*x* + 2 in for *y* into the second equation, then try to solve for *x*.

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

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

10*x* - 10*x* = 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.

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.

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.

Solution: We have our total bill and our friend's total bill. We don't know the cost per pound of apples and the cost per pound of strawberries. These are our two unknowns. Let *a* = the cost of 1 lb. of apples and *s* = the cost of 1 lb. of strawberries. You bought 2 pounds of apples, and the cost of each pound of apples is *a* dollars. Therefore you paid 2*a* dollars for apples. Similarly, you bought 1 pound of strawberries, and each pound of strawberries costs *s* dollars. Thus, you paid *s* dollars for strawberries. When you add these to costs together you get your total bill, which was $3.00, giving the equation 2*a* + *s* = 3. We use the same reasoning to find that your friend paid 6*a* dollars for apples and 3*s* dollars for strawberries for a total bill of $15.00. This gives the equation 6*a* + 3*s* = 15. We have our system of equations.

2*a* + *s* = 3

6*a* + 3*s* = 15

The solution to this system would represent the cost per pound of apples and strawberries if you and your friend paid the same amount per pound. Suppose we use elimination to solve this system.

This makes no sense. That is, 0 can't be equal to 6. This is an inconsistent system of equations, and there is no solution. Therefore, it is not possible that you and your friend paid the same amount per pound for the apples and strawberries. There must have been an error at the cash register, or a sale of some sort based on how many pounds were bought.

A **system of equations** is a collection of two or more equations that involve the same variables. When a system of equations has no solution, it is called **inconsistent**. If a system of equations is inconsistent, then when we try to solve it, we will end up with a statement that makes no sense, and if we observe the graphs of the equations involved, we will see that they never intersect. Being able to identify inconsistent equations is a great tool when it comes to determining if a certain scenario is possible.

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High School Algebra I: Tutoring Solution25 chapters | 257 lessons

- What is a Linear Equation? 7:28
- Applying the Distributive Property to Linear Equations 4:18
- Linear Equations: Intercepts, Standard Form and Graphing 6:38
- Abstract Algebraic Examples and Going from a Graph to a Rule 10:37
- Graphing Undefined Slope, Zero Slope and More 4:23
- Parallel, Perpendicular and Transverse Lines 6:06
- Graphs of Parallel and Perpendicular Lines in Linear Equations 6:07
- How to Write a Linear Equation 8:58
- What is a System of Equations? 8:39
- How Do I Use a System of Equations? 9:47
- Inconsistent Equation: Definition & Examples
- Go to High School Algebra - Linear Equations: Tutoring Solution

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