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Example 1 Solve
the following system of equations.
Solution
We are going to try and find values of x, y,
and a z that will satisfy all three
equations at the same time. We are
going to use elimination to eliminate one of the variables from one of the
equations and two of the variable from another of the equations. The reason for doing this will be apparent
once we’ve actually done it.
The elimination method in this case will work a little
differently than with two equations.
As with two equations we will multiply as many equations as we need to
so that if we start adding pairs of equations we can eliminate one of the
variables.
In this case it looks like if we multiply the second
equation by 2 it will be fairly simple to eliminate the y term from the second and third equation by adding the first
equation to both of them. So, let’s
first multiply the second equation by two.
Now, with this new system we will replace the second
equation with the sum of the first and second equations and we will replace
the third equation with the sum of the first and third equations.
Here is the resulting system of equations.
So, we’ve eliminated one of the variables from two of the
equations. We now need to eliminate
either x or z from either the second or third equations. Again, we will use elimination to do
this. In this case we will multiply
the third equation by -5 since this will allow us to eliminate z from this equation by adding the
second onto is.
Now, replace the third equation with the sum of the second
and third equation.
Now, at this point notice that the third equation can be
quickly solved to find that . Once we know this we can plug this into the
second equation and that will give us an equation that we can solve for z as follows.
Finally, we can substitute both x and z into the first
equation which we can use to solve for y. Here is that work.
So, the solution to this system is ,
and .
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