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SectionSSLESolving Systems of Linear Equations

We will motivate our study of linear algebra by considering the problem of solving several linear equations simultaneously. The word solve tends to get abused somewhat, as in “solve this problem.” When talking about equations we understand a more precise meaning: find all of the values of some variable quantities that make an equation, or several equations, simultaneously true.

SubsectionSLESystems of Linear Equations

Our first example is of a type we will not pursue further. While it has two equations, the first is not linear. So this is a good example to come back to later, especially after you have seen Theorem PSSLS.

In order to discuss systems of linear equations carefully, we need a precise definition. And before we do that, we will introduce our periodic discussions about “Proof Techniques.” Linear algebra is an excellent setting for learning how to read, understand and formulate proofs. But this is a difficult step in your development as a mathematician, so we have included a series of short essays containing advice and explanations to help you along. These will be referenced in the text as needed, and are also collected as a list you can consult when you want to return to re-read them. (Which is strongly encouraged!)

With a definition next, now is the time for the first of our proof techniques. So study Proof Technique D. We'll be right here when you get back. See you in a bit.

DefinitionSLESystem of Linear Equations

A system of linear equations is a collection of \(m\) equations in the variable quantities \(x_1,\,x_2,\,x_3,\ldots,x_n\) of the form, \begin{align*} a_{11}x_1+a_{12}x_2+a_{13}x_3+\dots+a_{1n}x_n&=b_1\\ a_{21}x_1+a_{22}x_2+a_{23}x_3+\dots+a_{2n}x_n&=b_2\\ a_{31}x_1+a_{32}x_2+a_{33}x_3+\dots+a_{3n}x_n&=b_3\\ &\vdots\\ a_{m1}x_1+a_{m2}x_2+a_{m3}x_3+\dots+a_{mn}x_n&=b_m \end{align*} where the values of \(a_{ij}\text{,}\) \(b_i\) and \(x_j\text{,}\) \(1\leq i\leq m\text{,}\) \(1\leq j\leq n\text{,}\) are from the set of complex numbers, \(\complexes\text{.}\)

Do not let the mention of the complex numbers, \(\complexes\text{,}\) rattle you. We will stick with real numbers exclusively for many more sections, and it will sometimes seem like we only work with integers! However, we want to leave the possibility of complex numbers open, and there will be occasions in subsequent sections where they are necessary. You can review the basic properties of complex numbers in Section CNO, but these facts will not be critical until we reach Section O.

Now we make the notion of a solution to a linear system precise.

DefinitionSSLESolution of a System of Linear Equations

A solution of a system of linear equations in \(n\) variables, \(\scalarlist{x}{n}\) (such as the system given in Definition SLE), is an ordered list of \(n\) complex numbers, \(\scalarlist{s}{n}\) such that if we substitute \(s_1\) for \(x_1\text{,}\) \(s_2\) for \(x_2\text{,}\) \(s_3\) for \(x_3\text{,}\) …, \(s_n\) for \(x_n\text{,}\) then for every equation of the system the left side will equal the right side, i.e. each equation is true simultaneously.

More typically, we will write a solution in a form like \(x_1=12\text{,}\) \(x_2=-7\text{,}\) \(x_3=2\) to mean that \(s_1=12\text{,}\) \(s_2=-7\text{,}\) \(s_3=2\) in the notation of Definition SSLE. To discuss all of the possible solutions to a system of linear equations, we now define the set of all solutions. (So Section SET is now applicable, and you may want to go and familiarize yourself with what is there.)

DefinitionSSSLESolution Set of a System of Linear Equations

The solution set of a linear system of equations is the set which contains every solution to the system, and nothing more.

Be aware that a solution set can be infinite, or there can be no solutions, in which case we write the solution set as the empty set, \(\emptyset=\set{}\) (Definition ES). Here is an example to illustrate using the notation introduced in Definition SLE and the notion of a solution (Definition SSLE).

We will often shorten the term system of linear equations to system of equations leaving the linear aspect implied. After all, this is a book about linear algebra.

SubsectionPSSPossibilities for Solution Sets

The next example illustrates the possibilities for the solution set of a system of linear equations. We will not be too formal here, and the necessary theorems to back up our claims will come in subsequent sections. So read for feeling and come back later to revisit this example.

This example exhibits all of the typical behaviors of a system of equations. A subsequent theorem will tell us that every system of linear equations has a solution set that is empty, contains a single solution, or contains infinitely many solutions (Theorem PSSLS). Example STNE yielded exactly two solutions, but this does not contradict the forthcoming theorem. The equations in Example STNE are not linear because they do not match the form of Definition SLE, and so we cannot apply Theorem PSSLS in this case.

SubsectionESEOEquivalent Systems and Equation Operations

With all this talk about finding solution sets for systems of linear equations, you might be ready to begin learning how to find these solution sets yourself. We begin with our first definition that takes a common word and gives it a very precise meaning in the context of systems of linear equations.

DefinitionESYSEquivalent Systems

Two systems of linear equations are equivalent if their solution sets are equal.

Notice here that the two systems of equations could look very different (i.e. not be equal), but still have equal solution sets, and we would then call the systems equivalent. Two linear equations in two variables might be plotted as two lines that intersect in a single point. A different system, with three equations in two variables might have a plot that is three lines, all intersecting at a common point, with this common point identical to the intersection point for the first system. By our definition, we could then say these two very different looking systems of equations are equivalent, since they have identical solution sets. It is really like a weaker form of equality, where we allow the systems to be different in some respects, but we use the term equivalent to highlight the situation when their solution sets are equal.

With this definition, we can begin to describe our strategy for solving linear systems. Given a system of linear equations that looks difficult to solve, we would like to have an equivalent system that is easy to solve. Since the systems will have equal solution sets, we can solve the “easy” system and get the solution set to the “difficult” system. Here come the tools for making this strategy viable.

DefinitionEOEquation Operations

Given a system of linear equations, the following three operations will transform the system into a different one, and each operation is known as an equation operation.

  1. Swap the locations of two equations in the list of equations.
  2. Multiply each term of an equation by a nonzero quantity.
  3. Multiply each term of one equation by some quantity, and add these terms to a second equation, on both sides of the equality. Leave the first equation the same after this operation, but replace the second equation by the new one.

These descriptions might seem a bit vague, but the proof or the examples that follow should make it clear what is meant by each. We will shortly prove a key theorem about equation operations and solutions to linear systems of equations.

We are about to give a rather involved proof, so a discussion about just what a theorem really is would be timely. Stop and read Proof Technique T first.

In the theorem we are about to prove, the conclusion is that two systems are equivalent. By Definition ESYS this translates to requiring that solution sets be equal for the two systems. So we are being asked to show that two sets are equal. How do we do this? Well, there is a very standard technique, and we will use it repeatedly through the course. If you have not done so already, head to Section SET and familiarize yourself with sets, their operations, and especially the notion of set equality, Definition SE, and the nearby discussion about its use.

The following theorem has a rather long proof. This chapter contains a few very necessary theorems like this, with proofs that you can safely skip on a first reading. You might come back to them later, when you are more comfortable with reading and studying proofs.


Theorem EOPSS is the necessary tool to complete our strategy for solving systems of equations. We will use equation operations to move from one system to another, all the while keeping the solution set the same. With the right sequence of operations, we will arrive at a simpler equation to solve. The next two examples illustrate this idea, while saving some of the details for later.

In the next section we will describe how to use equation operations to systematically solve any system of linear equations. But first, read one of our more important pieces of advice about speaking and writing mathematics. See Proof Technique L.

Before attacking the exercises in this section, it will be helpful to read some advice on getting started on the construction of a proof. See Proof Technique GS.

SubsectionReading Questions


How many solutions does the system of equations \(3x + 2y = 4\text{,}\) \(6x + 4y = 8\) have? Explain your answer.


How many solutions does the system of equations \(3x + 2y = 4\text{,}\) \(6x + 4y = -2\) have? Explain your answer.


What do we mean when we say mathematics is a language?



Find a solution to the system in Example IS where \(x_3=6\) and \(x_4=2\text{.}\) Find two other solutions to the system. Find a solution where \(x_1=-17\) and \(x_2=14\text{.}\) How many possible answers are there to each of these questions?


Each archetype (Appendix A) that is a system of equations begins by listing some specific solutions. Verify the specific solutions listed in the following archetypes by evaluating the system of equations with the solutions listed.

Archetype A, Archetype B, Archetype C, Archetype D, Archetype E, Archetype F, Archetype G, Archetype H, Archetype I, Archetype J


Find all solutions to the linear system. \begin{align*} x + y &= 5\\ 2x - y &= 3 \end{align*}


Find all solutions to the linear system. \begin{align*} 3x + 2y &= 1\\ x - y &= 2\\ 4x + 2y &= 2 \end{align*}


Find all solutions to the linear system. \begin{align*} x + 2y &= 8\\ x - y &= 2\\ x + y &= 4 \end{align*}


Find all solutions to the linear system. \begin{align*} x + y - z &= -1\\ x - y - z &= -1\\ z &= 2 \end{align*}


Find all solutions to the linear system. \begin{align*} x + y - z &= -5\\ x - y - z &= -3\\ x + y - z &= 0 \end{align*}


A three-digit number has two properties. The tens-digit and the ones-digit add up to 5. If the number is written with the digits in the reverse order, and then subtracted from the original number, the result is \(792\text{.}\) Use a system of equations to find all of the three-digit numbers with these properties.


Find all of the six-digit numbers in which the first digit is one less than the second, the third digit is half the second, the fourth digit is three times the third and the last two digits form a number that equals the sum of the fourth and fifth. The sum of all the digits is 24. (From The MENSA Puzzle Calendar for January 9, 2006.)


Driving along, Terry notices that the last four digits on his car's odometer are palindromic. A mile later, the last five digits are palindromic. After driving another mile, the middle four digits are palindromic. One more mile, and all six are palindromic. What was the odometer reading when Terry first looked at it? Form a linear system of equations that expresses the requirements of this puzzle. (Car Talk Puzzler, National Public Radio, Week of January 21, 2008) (A car odometer displays six digits and a sequence is a palindrome if it reads the same left-to-right as right-to-left.)


An article in The Economist (“Free Exchange”, December 6, 2014) quotes the following problem as an illustration that some of the “underlying assumptions of classical economics” about people's behavior are incorrect and “the mind plays tricks.” A bat and ball cost $1.10 between them. The bat costs $1 more than the ball. How much does each cost? Answer this quickly with no writing, then construct system of linear equations and solve the problem carefully.


Each sentence below has at least two meanings. Identify the source of the double meaning, and rewrite the sentence (at least twice) to clearly convey each meaning.

  1. They are baking potatoes.
  2. He bought many ripe pears and apricots.
  3. She likes his sculpture.
  4. I decided on the bus.

Discuss the difference in meaning of each of the following three almost identical sentences, which all have the same grammatical structure. (These are due to Keith Devlin.)

  1. She saw him in the park with a dog.
  2. She saw him in the park with a fountain.
  3. She saw him in the park with a telescope.

The following sentence, due to Noam Chomsky, has a correct grammatical structure, but is meaningless. Critique its faults. “Colorless green ideas sleep furiously.” (Chomsky, Noam. Syntactic Structures, The Hague/Paris: Mouton, 1957. p. 15.)


Read the following sentence and form a mental picture of the situation.

The baby cried and the mother picked it up.

What assumptions did you make about the situation?


Discuss the difference in meaning of the following two almost identical sentences, which have nearly identical grammatical structure. (This antanaclasis is often attributed to the comedian Groucho Marx, but has earlier roots.)

  • Time flies like an arrow.
  • Fruit flies like a banana.

This problem appears in a middle-school mathematics textbook: Together Dan and Diane have $20. Together Diane and Donna have $15. How much do the three of them have in total? (Transition Mathematics, Second Edition, Scott Foresman Addison Wesley, 1998. Problem 5–1.19.)


Solutions to the system in Example IS are given as \begin{equation*} (x_1,\,x_2,\,x_3,\,x_4)=(-1-2a+3b,\,4+a-2b,\,a,\,b) \end{equation*} Evaluate the three equations of the original system with these expressions in \(a\) and \(b\) and verify that each equation is true, no matter what values are chosen for \(a\) and \(b\text{.}\)


We have seen in this section that systems of linear equations have limited possibilities for solution sets, and we will shortly prove Theorem PSSLS that describes these possibilities exactly. This exercise will show that if we relax the requirement that our equations be linear, then the possibilities expand greatly. Consider a system of two equations in the two variables \(x\) and \(y\text{,}\) where the departure from linearity involves simply squaring the variables. \begin{align*} x^2-y^2&=1\\ x^2+y^2&=4 \end{align*} After solving this system of nonlinear equations, replace the second equation in turn by \(x^2+2x+y^2=3\text{,}\) \(x^2+y^2=1\text{,}\) \(x^2-4x+y^2=-3\text{,}\) \(-x^2+y^2=1\) and solve each resulting system of two equations in two variables. (This exercise includes suggestions from Don Kreher.)


Proof Technique D asks you to formulate a definition of what it means for a whole number to be odd. What is your definition? (Do not say “the opposite of even.”) Is \(6\) odd? Is \(11\) odd? Justify your answers by using your definition.


Explain why the second equation operation in Definition EO requires that the scalar be nonzero, while in the third equation operation this restriction on the scalar is not present.