Consistent System of Equations: Definition & Examples

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  • 0:03 System of Equations
  • 2:14 A Consistent System of…
  • 2:56 Examples & Non-Examples
  • 4:46 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we'll review what a system of equations is. We'll also define a specific type of system called a consistent system of equations, as well as look at some examples of systems to solidify our understanding of consistent systems.

System of Equations

A system of equations is a set of equations that have the same variables. For example, consider the set of the following two equations:

2x + y = 8

-4x - 3y = -20

This is a system of equations in x and y. The solution set of a system of equations is the set of all values of the variables that make each of the equations in the system true. For example, the solution set to the above system of equations is x = 2 and y = 4 because if we plug these values in for x and y in each of the equations, they make the equations true:

2(2) + 4 = 8

-4(2) - 3(4) = -20

The solution set can also be thought of as the set of intersection points on the graphs of the equations in the system. This image shows the graph of our system of equations. Notice their intersection point is (2,4):


To illustrate how these systems can be used, let's consider an example. Suppose you want to buy a blue or a red scarf. You can't decide which color you want, so you decide to just go with whichever one is cheaper. You know the retailer you want to buy the scarf from, but you don't know the price of each of the scarves. However, two of your friends have bought scarves there, and they have their receipts.

Your first friend bought 1 blue scarf and 1 red scarf and paid $35.00 total. Your other friend bought 2 blue scarves and 3 red scarves and paid $85.00. If we let b = the cost of a blue scarf and r = the cost of a red scarf, we can set up a system of equations to represent the situation:

b + r = 35

2b + 3r = 85

Notice that if we plug in 20 for b and 15 for r in each of the equations, both of the equations are true:

20 +15 = 35

2(20) + 3(15) = 85

Thus, (20,15) is the solution; that is, a blue scarf costs $20 and a red scarf costs $15. Thus, you decide you are going to buy the red scarf.

A Consistent System of Equations

When it comes to the solution of a system of equations, there are three possible outcomes:

  1. A finite number of solutions
  2. Infinitely many solutions
  3. No solution

Systems of equations can be placed into two categories: consistent and inconsistent. A consistent system of equations is a system that has at least one solution. An inconsistent system of equations is a system that has no solution. Thus, of the three possibilities for solutions of a system, we see that the first two possibilities represent consistent systems because they have at least one solution, and the third possibility represents an inconsistent system because it has no solution.

Examples and Non-Examples

Consider the following system of equations:

4x - y = 1

-8x + 2y = 4

Now,let's examine the graph of this system:


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