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Let’s start off this section with the definition of a linear equation. Here are a couple of examples of linear
equations.
In the second equation note the use of the subscripts on the
variables. This is a common notational
device that will be used fairly extensively here. It is especially useful when we get into the
general case(s) and we won’t know how many variables (often called unknowns)
there are in the equation.
So, just what makes these two equations linear? There are several main points to notice. First, the unknowns only appear to the first
power and there aren’t any unknowns in the denominator of a fraction. Also notice that there are no products and/or
quotients of unknowns. All of these
ideas are required in order for an equation to be a linear equation. Unknowns only occur in numerators, they are
only to the first power and there are no products or quotients of
unknowns.
The most general linear equation is,
where there are n
unknowns, 
,
and are all known numbers.
Next we need to take a look at the solution set of a single linear equation. A solution set (or often just solution) for (1)
is a set of numbers so that if we set ,
,
… , then (1)
will be satisfied. By satisfied we mean
that if we plug these numbers into the left side of (1) and
do the arithmetic we will get b as an
answer.
The first thing to notice about the solution set to a single
linear equation that contains at least two variables with non-zero coefficents
is that we will have an infinite number of solutions. We will also see that while there are
infinitely many possible solutions they are all related to each other in some
way.
Note that if there is one or less variables with non-zero
coefficients then there will be a single solution or no solutions depending
upon the value of b.
Let’s find the solution sets for the two linear equations
given at the start of this section.
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Example 1 Find
the solution set for each of the following linear equations.
(a) [Solution]
(b) [Solution]
Solution
(a)
The first thing that we’ll do here is solve the equation
for one of the two unknowns. It
doesn’t matter which one we solve for, but we’ll usually try to pick the one
that will mean the least amount (or at least simpler) work. In this case it will probably be slightly
easier to solve for so let’s do that.
Now, what this tells us is that if we have a value for then we can determine a corresponding value
for . Since we have a single linear equation
there is nothing to restrict our choice of and so we we’ll let be any number. We will usually write this as ,
where t is any number. Note that there is nothing special about
the t, this is just the letter that
I usually use in these cases. Others
often use s for this letter and, of
course, you could choose it to be just about anything as long as it’s not a
letter representing one of the unknowns in the equation (x in this case).
Once we’ve “chosen” we’ll write the general solution set as
follows,
So, just what does this tell us as far as actual number
solutions go? We’ll choose any value
of t and plug in to get a pair of
numbers and that will satisfy the equation. For instance picking a couple of values of t completely at random gives,
We can easily check that these are in fact solutions to
the equation by plugging them back into the equation.
So, for each case when we plugged in the values we got for
and we got -1 out of the equation as we were
supposed to.
Note that since there an infinite number of choices for t there are in fact an infinite number
of possible solutions to this linear equation.
[Return to Problems]
(b)
We’ll do this one with a little less detail since it works
in essentially the same manner. The
fact that we now have three unknowns will change things slightly but not
overly much. We will first solve the
equation for one of the variables and again it won’t matter which one we
chose to solve for.
In this case we will need to know values for both x and y in order to get a value for z. As with the first case, there is nothing in
this problem to restrict out choices of x
and y. We can therefore let them be any
number(s). In this case we’ll choose and . Note that we chose different letters here
since there is no reason to think that both x and y will have
exactly the same value (although it is
possible for them to have the same value).
The solution set to this linear equation is then,
So, if we choose any values for t and s we can get a
set of number solutions as follows.
As with the first part if we take either set of three
numbers we can plug them into the equation to verify that the equation will
be satisfied. We’ll do one of them and
leave the other to you to check.
[Return to Problems]
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The variables that we got to
choose for values for (
in the first example and x and y in the second) are sometimes called free variables.
We now need to start talking about the actual topic of this
section, systems of linear equations. A system of linear equations is nothing
more than a collection of two or more linear equations. Here are some examples of systems of linear
equations.
As we can see from these examples systems of equation can
have any number of equations and/or unknowns.
The system may have the same number of equations as unknowns, more
equations than unknowns, or fewer equations than unknowns.
A solution set to a system with n unknowns, ,
is a set of numbers, ,
so that if we set ,
,
… , then all of the equations in the system will
be satisfied. Or, in other words, the
set of numbers is a solution to each of the individual
equations in the system.
For example, ,
is a solution to the first system listed
above,
because,
However, ,
is not a solution to the system because,
We can see from these calculations that ,
is NOT a solution to the first equation, but
it IS a solution to the second equation.
Since this pair of numbers is not a solution to both of the equations in
(2)
it is not a solution to the system. The
fact that it’s a solution to one of them isn’t material. In order to be a solution to the system the
set of numbers must be a solution to each and every equation in the system.
It is completely possible as well that a system will not
have a solution at all. Consider the
following system.