Solving Simultaneous Linear Equations

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  • 0:01 Simultaneous Linear Equations
  • 0:59 Gaussian Elimination
  • 1:32 Eliminating our…
  • 3:06 Finding the Solutions
  • 4:40 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, you will learn how to solve simultaneous linear equations or a system of linear equations. Learn the steps you need to take to solve any such system using the Gaussian elimination method.

Simultaneous Linear Equations

If one runner in a race represents one linear equation, then several runners in the same race represents a group of simultaneous linear equations, since they are all running the same race at the same time.

We can define simultaneous linear equations as a collection of linear equations. In math, we also call it a system of linear equations. Actually, the term that you are most likely to see is the system of linear equations. The number of equations we have in our system of linear equations depends on the number of variables we have.

For example, if we have three variables, then we have three equations. Just like with our runners, if we have three different names, then we have three different runners. Also, each variable has no exponent attached to it, just like each name represents just one runner. In this video lesson, we are going to learn how to solve such a system. Let's practice solving this system:

linear system

Gaussian Elimination

The method that we will be using is called the Gaussian elimination method. This method requires us to eliminate the first variable in the second equation and then the first two variables in the third equation and so on, if we have more variables, until we reach the last equation where we are left with the last variable.

To eliminate our variables, we are going to add two equations together so we can eliminate one of the variables. We repeat this process until we have eliminated all the variables that we wanted to. Let's see how this is done.

Eliminating Our Unwanted Variables

We can leave the first equation alone, since there are no variables that we want to eliminate here. In our second equation, we want to eliminate the x variable. In our third equation, we want to eliminate our y variable. Our x variable in our third equation is already gone, so we don't need to worry about eliminating that one.

To eliminate the x variable in our second equation, we look at all our equations, and we see that we can add the first and second equation together to eliminate the x variable here. Doing this, we get (2x + y - 3z = - 5) + (-2x + y + z = 3) = (2y - 2z = -2). We can now replace our second equation with this new equation. Our system now looks like this:

linear system

Next, we need to eliminate the y in the third equation. Looking at all our equations again, we now see that if we multiply our third equation by -2 and then add it to the second equation, we can eliminate our y variable. Doing this, we get (2y - 2z = -2) + (-2)(y + z = 5) = (2y - 2z = -2) + (-2y - 2z = -10) = (-4z = -12). This new equation is now our third equation. Our system now looks like this:

linear system

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