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Amy has a master's degree in secondary education and has taught math at a public charter high school.
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:
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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.
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:
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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:
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Now that we have eliminated all the variables that we wanted to, we can now go ahead and solve our system. To do this, we begin with our last equation to solve for our last variable. We divide our last equation by -4 to get z by itself. We find that z = 3.
Now we can plug this value for z into the second equation and then solve this second equation for y. We get 2y - 2(3) = -2, which becomes 2y - 6 = -2. Adding 6 to both sides, we get 2y = 4. Dividing by 2, we get y = 2.
Now that we have both y and z, we can now plug these into our first equation to solve for our first variable. We have 2x + 2 - 3(3) = -5. This becomes 2x + 2 - 9 = -5. Combining like terms, we get 2x - 7 = -5. Adding 7 to both sides, we get 2x = 2. Dividing by 2, we get x = 1.
So, our complete solution is x = 1, y = 2, and z = 3. We can also write it in point form as (1, 2, 3), where our variables appear in alphabetical order. Do you see how we essentially worked our way back up the first equation to solve our system?
Let's review what we've learned now. We can define simultaneous linear equations as a collection of linear equations. In math, we also call it a system of linear equations. To solve such a system, we can use 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. Once we have eliminated all the variables that we want, then we can start with our last equation to solve for the last variable. Then we work our way up to our first equation, plugging in our variables as we find them and solving for the next unknown variable. We keep going until we have solved all the variables.
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