Back To CourseAlgebra II: High School
23 chapters | 203 lessons
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
You are probably familiar with linear equations, such as y = 3x + 4 or y - 3x = 4. These equations, when graphed, will give you a straight line. A linear system, then, is a collection of linear equations. Our usual linear equations have only two variables. Because we have two variables, a linear system using these kinds of equations will have two equations.
But, did you know that you can also have a linear system with three variables? Yes, you can. A linear system in three variables, then, will have three equations since it has three variables. These equations, just like the ones with only two variables, will not have any exponents. This is an example of a linear system in three variables:
Looking at this system, we see our three variables, x, y, and z. These three letters are the most commonly used, although you can use any letter that you want; they don't have to be in order, either. If you have a friend named Sam, you could easily use those three letters, too. The important thing to remember is that we have three different variables, and none of them have exponents with them.
You will see these kinds of linear systems in higher math, where you will be asked to solve them. Just like other systems, our linear system in three variables can have one solution, no solution, or an infinite number of solutions. Just like the other equations that we are used to solving, if our system has only one solution, then we look for just one point. But, what about the other two? How do you find out if our system has no solution or an infinite number of solutions? Let's find out.
Each equation in our system, when graphed, produces a plane - a flat surface that goes on forever. Because we are dealing with three variables, we are dealing with 3-dimensional space. So, picture the plane floating in space. We have three equations, so we have three of these flat surfaces floating around.
To simulate this on a smaller scale, simply take three pieces of paper, have a friend lend you a third hand, and hold onto them in random places in front of you. Imagine that these sheets of paper don't end; they keep going. As you keep playing, you will notice that many times, your three sheets of paper don't all meet together.
In this case, you will have no solution. Even if two of the planes meet, if the third one doesn't meet at the same point or points as the other two, then there is no solution. How does this look mathematically? Let's see. Let's try and solve this system:
We can use any method that we are comfortable with to try and solve for our variables. Since we have a nicely laid out system, I'm going to use the elimination method to eliminate some of the variables to make it easier for me to solve.
I look at the first and third equations and see that they are ready for me to combine. If I add these two equations, I can make at least the x variable disappear, meaning I can make the coefficient of the x variable 0. I add these two equations together. I get 0 + 0 + 0 = 4.
Hmmm… does this make sense? No, it doesn't. 0 can never equal 4. I can stop right there. Since the third and first equations produced a false statement, that tells me that this system has no solution. For a linear system in three variables to have no solution, all you need is to find two equations, that when combined, produce a false statement.
What about infinite solutions? We have two cases where we end up with an infinite number of solutions. The first is when the planes are all lined up on the same plane, and the second is when they all intersect one line. For this lesson, we focus our attention on the first case, when all the planes or sheets of paper all lie on the same plane. Let's see what this looks like mathematically:
We will use the elimination method again to try and solve this. We first combine the first and third equation by adding them together, since we can make at least the x variable disappear. We get 0 + 0 + 0 = 0, which turns into 0 = 0. That's a true statement. It tells me that the first and third equation are the same plane.
What about the second equation? Let's see. I can multiply the third equation by 3 and then add it to the second to make at least one of the variables disappear. Multiplying the third equation by 3, I get -3x - 3y - 3z = -9.
Adding this to the second equation, I get 0 + 0 + 0 = 0. Is this a true statement? 0 = 0. Yes, it's a true statement; that means all three of these equations are on the same plane. We can, therefore, say that this system has an infinite number of solutions.
Let's review what we've learned now. We learned that a linear system is a collection of linear equations. If we have a linear system in three variables, then our equations will each have three variables, and we will have a total of three equations.
To solve these systems of equations, we can use the elimination method, which involves possibly multiplying one equation with a number and then adding it to another equation in the system to try and eliminate some of the variables
If our system has no solutions, then two of the equations will produce a false statement, such as 3 = 4. If our system has an infinite number of solutions, then all three of the equations, when combined with each other, will produce a true statement, such as 0 = 0.
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Back To CourseAlgebra II: High School
23 chapters | 203 lessons