Example 2 Solve
the following system of equations.
Solution
Before we get started on the solution process do not get
excited about the fact that the second equation only has two variables in
it. That is a fairly common occurrence
when we have more than two equations in the system.
In fact, we’re going to take advantage of the fact that it
only has two variables and one of them, the y, has a coefficient of 1.
This equation is easily solved for y
to get,
We can then
substitute this into the first and third equation as follows,
Now, if you
think about it, this is just a system of two linear equations with two
variables (x and z) and we know how to solve these
kinds of systems from our work in the previous section.
First, we’ll
need to do a little simplification of the system.
The simplified
version looks just like the systems we were solving in the previous
section. Well, it’s almost the
same. The variables this time are x and z instead of x and y, but that really isn’t a
difference. The work of solving this
will be the same.
We can use
either the method of substitution or the method of elimination to solve this
new system of two linear equations.
If we wanted to
use the method of substitution we could easily solve the second equation for z (you do see why it would be easiest
to solve the second equation for z
right?) and substitute that into the first equation. This would allow us to find x and we could then find both z and y.
However, to
make the point that often we use both methods in solving systems of three
linear equations let’s use the method of elimination to solve the system of
two equations. We’ll just need to
multiply the first equation by 3 and the second by 5. Doing this gives,
We can now
easily solve for x to get . The coefficients on the second equation are
smaller so let’s plug this into that equation and solve for z.
Here is that work.
Finally, we
need to determine the value of y. This is very easy to do. Recall in the first step we used
substitution and in that step we used the following equation.
Since we know
the value of x all we need to do is
plug that into this equation and get the value of y.
Note that in
many cases where we used substitution on the very first step the equation
you’ll have at this step will contain both x’s and z’s and so you
will need both values to get the third variable.
Okay, to finish this example up here is the solution : ,
and .
