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Online Notes / Linear Algebra / Systems of Equations and Matrices / Systems of Equations
Linear Algebra

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 Systems of Equations

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,

(1)

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.

 

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]

 

 

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,

(2)

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.