SE - Systems of Equations Lesson

Systems of Equations

What exactly is a system of equations? Well, a system of equations is two or more equations that show a relationship among the given variables. For now, we'll only focus on systems that have two equations. Just know that the system can have more than two equations.

Here is a system of equations:

 

We can solve a system by several methods. The methods we'll discuss are as follows:

• Graphing Method
• Substitution Method
• Elimination Method

What exactly are we finding when we solve a system of equations? Well, when we solve a system, we are looking for that one point, or ordered pair, that makes both equations true. Graphically, that means that we are looking for the point where the lines intersect.

In most instances, we will run across situations where there is only one solution....that one point where the graphs intersect. But, just like when we are solving equations, there are those situations where we get a "no solutions" or an "infinitely many solutions" answer. How would that happen with a system? Well, let's think. What is the only time you can have two lines that would never touch or intersect? If you said when they are parallel, that's exactly right!!

If you have parallel lines, then there is no solution to the system because those two lines will never, ever intersect, or share a point. This means no ordered pair will ever satisfy both equations.

Now, let's investigate the other special situation of having infinitely many solutions. If the system has infinitely many solutions, then they share infinitely many ordered pairs. How might that happen? Well, it would happen if the lines are co-linear. In other words, if they are the same line, they share each of their points.

Let's see an example of each way of solving a system. The following 3 videos are from Khan Academy. Once you finish the first video, be sure to click on the arrow at the top right of the video player in order to move on to the next one.

Want more examples? Click on the following for additional information:

• Graphing Method Links to an external site.
• Substitution Method Links to an external site.
• Elimination Method Links to an external site.

Now that we know how to solve systems using these three methods, it is important for us to be able to apply this knowledge to real world situations.

Let's first watch a Khan Academy Links to an external site. example of an application problem using systems:

EXAMPLE:
Maggie is hosting a party after school for the students at the middle school. The principal has given her two guidelines that she must follow. First, the total number of people attending (teachers and students combined) must be 56. Secondly, there must be one teacher for every 7 students. How many students and how many teachers are invited to the party?

First, we need to identify our variables.
s = students
t = teachers

Since the total number of students and teachers has to be 56, that will be our first equation:

Our other information will help us to set up our second equation. We know that for every 7 students, we must have one teacher. We can use the follow equation to represent that relationship:

 

 

Now, we have the two equations to complete our system:

We can use any method discussed above to solve this system. 

Assignment

Now that you have spent some time learning about systems of equations, you are ready to complete your Systems of Equations Homework. Click here to download the Systems of Equations Homework Links to an external site..

Once you have completed the Systems of Equations Homework, check your even answers and make sure you ask your teacher if you have any questions Links to an external site.. After you complete your Systems of Equations HW and view the feedback, consulting your teacher on any you do not understand, you're ready to take the Systems of Equations Quiz.

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