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Solving a System of Equations with No Solution

Solving a System of Equations with No Solution
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  • 0:05 A System of Equations…
  • 1:58 Graphing
  • 3:03 Logic
  • 4:08 Algebra
  • 5:40 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

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

This lesson will enable us to determine if a system of equations has no solution. We will look at three different ways to go about solving a system with no solution, and we will explain what happens in each situation.

A System of Equations With No Solution

Suppose you're going to run a lemonade and cookie stand. You're trying to decide how to price each item, and you want to know if it's possible to price them such that 1 lemonade and 2 cookies costs $4.00 and 3 lemonades and 6 cookies cost $18.00.

You decide to use math to figure this out. You represent the cost of one lemonade with the variable x and the cost of one cookie with the variable y. Using these variables, we represent the first scenario with the equation x + 2y = 4, because we have one lemonade at x dollars and 2 cookies at y dollars each. Adding the cost of these together gives $4.00. Similarly, we represent the second scenario with 3x + 6y = 18. Thus, you have the two following equations:

x + 2y = 4

3x + 6y = 18

This set of equations is called a system of equations. A system of equations is a set of equations in the same variables. The solution set to a system of equations is the set of all values of the variables that make all of the equations in the system true. The solution set can also be thought of as the set of all intersection points of the graphs of each of the equations in the set. Thus, in our lemonade stand problem, you want to know if this system of equations has a solution.

Spoiler alert: it doesn't.

I'm telling you this now, because we're going to look at trying to solve systems with no solution, and this is a great example to use to illustrate the different ways of trying to solve these types of systems.

An inconsistent system of equations is a system of equations that has no solution. Consider our example. This system has no solution, so we would say that it's inconsistent. We're going to look at three different ways to identify inconsistent systems, and we'll use our example to illustrate each of these.

Graphing

The solution set of a system of equations can be thought of as the set of all the intersection points of the graphs of the equations in the system. If the system has no solution, then there are no points of intersection of the graphs of the equations in the system, so the graphs of the equations must never intersect. Thus, if we graph all the equations in our system, and observe that they never intersect, then we know we have an inconsistent system. Let's look at our example. We graph both of the equations on the same graph as shown:


Graph of our System
1


We see that the graphs are parallel lines, thus they never intersect, so we know the system is inconsistent and has no solution.

It is important to note that sometimes graphs can be misleading. For instance, consider this system of equations and their graphs:


Lines Appear Parallel
2


It appears that the two graphs do not intersect; however, if you zoom out a bit or look at a different part of the graph, you would find that they would eventually intersect at the point (20,25).


We see the lines are not parallel
3


Therefore, it is always important to either use algebra or logic, along with graphing, to make sure the system is inconsistent.

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