Explainer: Solution Cases of System of Linear Equations

In this explainer, we will learn how to determine whether a linear system of equations has a unique solution, no solution, or an infinite number of solutions.

When working with a system of linear equations, it is easy to assume that there is a solution. This solution might be a unique solution, where all of the variables have only one possible value. Alternatively, there may be infinitely many solutions, meaning that each variable can be expressed in terms of some independent parameters. Although there are fundamental differences in these two types of solutions, they are similar to each other in the sense that they represent systems of linear equations which do actually have a solution. We broadly categorize these two solution types by saying that they correspond to systems of linear equations that are โ€œconsistent.โ€ It may also be the case that it is not possible to obtain a solution to a system of linear equations, in which case the problem is known as โ€œinconsistent.โ€ For example, if we were to take the system of linear equations ๐‘ฅ+๐‘ฆ=โˆ’3,2๐‘ฅ+2๐‘ฆ=โˆ’8, then we would quickly find that it is not possible to solve this problem. Thinking in a geometric sense, it is because the two lines encoded by the two equations are parallel, meaning that they will never intersect. In this case we would say that the system is โ€œinconsistent,โ€ as it is not possible to find a solution. This can be more clearly shown by rearranging both of the above equations for ๐‘ฆ to read ๐‘ฆ=โˆ’3โˆ’๐‘ฅ,๐‘ฆ=โˆ’4โˆ’๐‘ฅ.

Now we have ๐‘ฆ as the subject of both equations, and we can equate them to find โˆ’3โˆ’๐‘ฅ=โˆ’4โˆ’๐‘ฅ, which cancels to give โˆ’3=โˆ’4, which is blatantly absurd. Given this logical inconsistency, we say that the whole system of linear equations is inconsistent.

On the basis of this preliminary example, it is probably worth asking ourselves what we might expect to see if we are working with an inconsistent system of linear equations. Suppose we take, for example, the system of linear equations ๐‘ฅ+๐‘ฆ+2๐‘ง=โˆ’7,2๐‘ฅโˆ’๐‘ฆโˆ’3๐‘ง=8,4๐‘ฅโˆ’2๐‘ง=4.

It is not obvious, at this stage, whether the system of linear equations has a unique solution, has infinitely many solutions, or does not have any solutions at all. In this instance, any appropriate method would show that there is a unique solution with the values ๐‘ฅ=โˆ’1,๐‘ฆ=2,๐‘ง=โˆ’4.

We have formatted the answer in a very particular way, the reasons for which will become apparent later on in the explainer. Now suppose that, instead of the above set of equations, we had begun with a system of linear equations which led us to the values ๐‘ฅ=โˆ’1,๐‘ฆ=2,0ร—๐‘ง=โˆ’4.

The last line is obviously ridiculous, as it suggests that 0=โˆ’4. In this case, we would have said that the system of linear equations was inconsistent, as it produces an absurdity. We will now take the time to properly define what it means for a system of equations to be inconsistent, before preparing a second definition that will allow us to more easily classify systems of equations as either consistent or inconsistent.

Definition: Systems of Linear Equations Which Are Inconsistent

Consider the system of linear equations in the variables ๐‘ฅ,๐‘ฅ,โ€ฆ,๐‘ฅ๏Šง๏Šจ๏Š: ๐‘Ž๐‘ฅ+๐‘Ž๐‘ฅโ‹ฏ๐‘Ž๐‘ฅ=๐‘,๐‘Ž๐‘ฅ+๐‘Ž๐‘ฅโ‹ฏ๐‘Ž๐‘ฅ=๐‘,โ‹ฎโ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎโ‹ฎ๐‘Ž๐‘ฅ+๐‘Ž๐‘ฅโ‹ฏ๐‘Ž๐‘ฅ=๐‘.๏Šง๏Šง๏Šง๏Šง๏Šจ๏Šจ๏Šง๏Š๏Š๏Šง๏Šจ๏Šง๏Šง๏Šจ๏Šจ๏Šจ๏Šจ๏Š๏Š๏Šจ๏‰๏Šง๏Šง๏‰๏Šจ๏Šจ๏‰๏Š๏Š๏‰

Then the system is said to be โ€œconsistentโ€ if there exists at least one set of values for ๐‘ฅ,๐‘ฅ,โ€ฆ,๐‘ฅ๏Šง๏Šจ๏Š which solve all of the equations. Otherwise, the system is called โ€œinconsistentโ€.

When attempting to solve a system of linear equations, the most common method is arguably the Gaussโ€“Jordan method, where elementary row operations are used to manipulate the system of equations into an expression for the solution, without changing any properties of the system itself. Most often, this will be achieved by first writing the system of equations in a convenient form which encapsulates all of the information, which in this case is the corresponding augmented coefficient matrix.

Definition: Augmented Coefficient Matrix

Consider a general system of linear equations in the variables ๐‘ฅ,๐‘ฅ,โ€ฆ,๐‘ฅ๏Šง๏Šจ๏‰ and the coefficients ๐‘Ž๏ƒ๏…: ๐‘Ž๐‘ฅ+๐‘Ž๐‘ฅโ‹ฏ๐‘Ž๐‘ฅ=๐‘,๐‘Ž๐‘ฅ+๐‘Ž๐‘ฅโ‹ฏ๐‘Ž๐‘ฅ=๐‘,โ‹ฎโ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎโ‹ฎ๐‘Ž๐‘ฅ+๐‘Ž๐‘ฅโ‹ฏ๐‘Ž๐‘ฅ=๐‘.๏Šง๏Šง๏Šง๏Šง๏Šจ๏Šจ๏Šง๏Š๏Š๏Šง๏Šจ๏Šง๏Šง๏Šจ๏Šจ๏Šจ๏Šจ๏Š๏Š๏Šจ๏‰๏Šง๏Šง๏‰๏Šจ๏Šจ๏‰๏Š๏Š๏‰

The โ€œaugmented coefficient matrixโ€ of the system is โŽกโŽขโŽขโŽขโŽขโŽฃ๐‘Ž๐‘Žโ‹ฏ๐‘Ž๐‘๐‘Ž๐‘Žโ‹ฏ๐‘Ž๐‘โ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎ๐‘Ž๐‘Žโ‹ฏ๐‘Ž๐‘โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ.๏Šง๏Šง๏Šง๏Šจ๏Šง๏Š๏Šง๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏Š๏Šจ๏‰๏Šง๏‰๏Šจ๏‰๏Š๏‰

In translating from a system of linear equations to its augmented coefficient matrix, we have essentially already understood one of the key steps for determining whether the system is consistent or inconsistent. To elucidate this method, we take the augmented coefficient matrix of a system of linear equations where there are ๐‘› equations with ๐‘› variables, writing these variables above the columns that they correspond to:

Suppose now that we used elementary row operations, or any other suitable method, to transform this system into the reduced echelon form

such that every entry is zero except for the diagonal entries, which all have a value of 1, and the ฬ‚๐‘๏ƒ entries, which can take any value. Then, writing this coefficient matrix as a system of linear equations gives ๐‘ฅ=ฬ‚๐‘,๐‘ฅ=ฬ‚๐‘,โ‹ฑโ‹ฎโ‹ฎ๐‘ฅ=ฬ‚๐‘.๏Šง๏Šง๏Šจ๏Šจ๏Š๏Š

This corresponds to a unique solution, where ๐‘ฅ,๐‘ฅ,โ€ฆ,๐‘ฅ๏Šง๏Šจ๏Š may only take one possible value in order to solve the system of linear equations. In this case the system is consistent, as there is at least one valid solution. However, suppose we had arrived at the reduced echelon form

where the only difference is in the highlighted entry. Supposing that ฬ‚๐‘โ‰ 0๏Š, then writing out this augmented matrix in terms of a system of linear equations gives ๐‘ฅ=ฬ‚๐‘,๐‘ฅ=ฬ‚๐‘,โ‹ฑโ‹ฎโ‹ฎ0=ฬ‚๐‘.๏Šง๏Šง๏Šจ๏Šจ๏Š

This clearly does not make any sense, as we cannot have 0โ‰ ฬ‚๐‘๏Š if the latter quantity is nonzero. This would, therefore, represent a system that is inconsistent. The property is summarized by the following theorem.

Theorem: Inconsistency of a System of Linear Equations

Suppose that a system of equations is written into the corresponding augmented coefficient matrix. If this matrix is transformed into to an echelon form and there is a pivot in the right-most column, then the system is inconsistent. If there are no pivots in the right-most column then the system of linear equations is consistent.

A pivot is the first nonzero entry of a matrix. For an augmented coefficient matrix, if one of the pivots is in the right-most column, then all other entries in the row are zero. For variables ๐‘ฅ,๐‘ฅ,โ€ฆ,๐‘ฅ๏Šง๏Šจ๏Š, suppose this applied to row ๐‘— of the augmented coefficient matrix after some number of row operations had been performed. By having the first nonzero entry in the right-most column, this would imply that 0ร—๐‘ฅ+0ร—๐‘ฅ+โ‹ฏ+0ร—๐‘ฅ=ฬ‚๐‘,๏Šง๏Šจ๏Š๏… where ฬ‚๐‘โ‰ 0๏…. This is clearly impossible as the left-hand side is equal to zero and the right-hand side is not. We will demonstrate one example of this property in practice.

Suppose that we have the system of linear equations ๐‘ฅโˆ’2๐‘ฆโˆ’๐‘ง=3,โˆ’3๐‘ฅ+11๐‘ฆ+5๐‘ง=โˆ’7,7๐‘ฅโˆ’24๐‘ฆโˆ’11๐‘ฆ=25.

This can be represented by the augmented coefficient matrix โŽกโŽขโŽขโŽฃ1โˆ’2โˆ’13โˆ’3115โˆ’77โˆ’24โˆ’1125โŽคโŽฅโŽฅโŽฆ.

We attempt to solve this system using the method of Gaussโ€“Jordan elimination, by using elementary row operations to attempt to manipulate the matrix into reduced echelon form. We first highlight all of the pivot entries: โŽกโŽขโŽขโŽฃ1โˆ’2โˆ’13โˆ’3115โˆ’77โˆ’24โˆ’1125โŽคโŽฅโŽฅโŽฆ.

To begin working towards reduced echelon form, we must eliminate all of the nonzero entries which are below the pivot in the first row. We use the joint row operations ๐‘Ÿโ†’๐‘Ÿ+3๐‘Ÿ๏Šจ๏Šจ๏Šง and ๐‘Ÿโ†’๐‘Ÿโˆ’7๐‘Ÿ๏Šฉ๏Šฉ๏Šง to give the matrix โŽกโŽขโŽขโŽฃ1โˆ’2โˆ’1305220โˆ’10โˆ’44โŽคโŽฅโŽฅโŽฆ.

Now we need to eliminate the nonzero entry which is below the pivot in the second row. This is achieved with the row operation ๐‘Ÿโ†’๐‘Ÿ+2๐‘Ÿ๏Šฉ๏Šฉ๏Šจ, giving the matrix โŽกโŽขโŽขโŽฃ1โˆ’2โˆ’1305220008โŽคโŽฅโŽฅโŽฆ.

This matrix is in an echelon form and yet there is a pivot in the right-most column. By the theorem above, this means that the system is inconsistent. Writing out this system of equations would give ๐‘ฅโˆ’2๐‘ฆโˆ’๐‘ง=3,5๐‘ฆ+2๐‘ง=2,0ร—๐‘ง=8.

The third equation contains a false statement, validating our assertion that the system is inconsistent.

We will now give 3 examples where we must recognize whether or not a given augmented matrix is representing a system of linear equations that is consistent, or inconsistent. If the system of equations is consistent, then we will also be expected to recognize whether or not the system of equations has a unique solution or infinitely many solutions.

Example 1: Recognizing an Inconsistent System of Equations

In the given augmented matrix, โˆ— denotes an arbitrary number and โ–  denotes an arbitrary nonzero number. Determine whether the given augmented matrix is consistent and, if it is consistent, whether its solution is unique: โŽกโŽขโŽขโŽขโŽขโŽฃโ– โˆ—โˆ—โˆ—โˆ—โˆ—0โ– โˆ—โˆ—โˆ—โˆ—00โ– โˆ—โˆ—โˆ—0000โ– โˆ—โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ.

Answer

This matrix is in an echelon form. Given that the โ–  entries are nonzero, they actually represent the pivots of each row. Since the matrix is in an echelon form and there are no pivots in the right-most column, the system of equations is consistent.

There are 5 variables in this system of equations and yet there are only 4 equations overall. Given that there are more variables than equations, it is not possible for the solution to be unique and therefore there must be infinitely many solutions.

Example 2: Recognizing an Inconsistent System of Equations

In the augmented matrix โŽกโŽขโŽขโŽฃโ– โˆ—โˆ—โˆ—0โ– โˆ—โˆ—00โ– โˆ—โŽคโŽฅโŽฅโŽฆ,โˆ— denotes an arbitrary number and โ–  denotes a nonzero number. Determine whether the given augmented matrix is consistent. If it is consistent, is the solution unique?

Answer

The augmented matrix is in an echelon form and there are no pivot entries in the right-most column, which means that the corresponding system of equations is consistent.

There are 3 variables in this system and, with 3 equations containing them, this means that the solution must be unique.

Example 3: Recognizing an Inconsistent System of Equations

In the augmented matrix โŽกโŽขโŽขโŽขโŽขโŽฃโ– โˆ—โˆ—โˆ—โˆ—โˆ—0โ– โˆ—โˆ—0โˆ—0000โ– 00000โˆ—โ– โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ,โˆ— denotes an arbitrary number and โ–  denotes an arbitrary nonzero number. Determine whether the given augmented matrix is consistent and, if it is consistent, whether its solution is unique.

Answer

The key to this question is in the highlighted entry: โŽกโŽขโŽขโŽขโŽขโŽฃโ– โˆ—โˆ—โˆ—โˆ—โˆ—0โ– โˆ—โˆ—0โˆ—0000โ– 00000โˆ—โ– โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ.

If this entry is nonzero, then it is a pivot entry. Given that the augmented matrix is already in an echelon form, if the โˆ— entry is nonzero, then this would be a pivot entry and therefore the solution would be consistent. Rather than the solution being unique, there would be infinitely many solutions because there are fewer equations than there are variables in the corresponding system.

If the โˆ— entry is zero, then there will be a pivot entry in the right-most column, which means that the system is inconsistent.

Since the last entry of the third row is zero, then the 4th variable in the same row must be zero. This leads to the โˆ— entry being zero, then there will be a pivot entry in the right-most column, which means that the system is inconsistent.

The previous questions were based on recognizing whether an augmented matrix is consistent, by looking at the entries and the location of the pivots. In the two questions below we will examine specific augmented matrices containing unknown variables, and we will use row operations to show how these systems of linear equations can be classified with respect to these variables.

Example 4: Calculating When a Systems of Linear Equations Is Inconsistent

Find the values of โ„Ž for which the augmented matrix ๏˜2โ„Ž4367๏ค is consistent.

Answer

We will begin by maneuvering the matrix into an echelon form. To do this, we first highlight the pivots of the given matrix: ๏˜2โ„Ž4367๏ค.

To eliminate the nonzero entry that is below the pivot in the first row, we use the row operation ๐‘Ÿโ†’๐‘Ÿโˆ’32๐‘Ÿ๏Šจ๏Šจ๏Šง, which gives ๏™2โ„Ž406โˆ’32โ„Ž1๏ฅ.

This matrix is now in an echelon form. To be a pivot, the highlighted entry in the second row must be nonzero, as otherwise the matrix would take the form ๏˜2โ„Ž4001๏ค.

This matrix has a pivot in the right-most column and is therefore inconsistent. To avoid this, we must have 6โˆ’32โ„Žโ‰ 0, which means that โ„Žโ‰ 4. For any other value of โ„Ž, the system will be consistent. Due to the fact that there is the same number of variables as there are equations, the corresponding system would have a unique solution.

Example 5: Calculating When a Systems of Linear Equations Is Inconsistent

Find conditions on โ„Ž and ๐‘˜ for the following augmented matrix to have no solution, a unique solution, and infinitely many solutions: ๏˜1222โ„Ž๐‘˜๏ค.

Answer

We highlight the pivot entries of the given augmented coefficient matrix: ๏˜1222โ„Ž๐‘˜๏ค.

To change this matrix into an echelon form, we must first eliminate the nonzero entry which is below the pivot in the first row. We use the row operation ๐‘Ÿโ†’๐‘Ÿโˆ’2๐‘Ÿ๏Šจ๏Šจ๏Šง to give ๏˜1220โ„Žโˆ’4๐‘˜โˆ’4๏ค.

If โ„Žโˆ’4โ‰ 0, then the corresponding system of equations will be consistent. Given that there is the same number of equations as there are variables, the solution must therefore be unique.

If โ„Žโˆ’4=0, then we have the matrix ๏˜12200๐‘˜โˆ’4๏ค and there are two possibilities. One is that ๐‘˜โˆ’4โ‰ 0 and there is a pivot in the right-most column, meaning that the solution is inconsistent. Alternatively, ๐‘˜โˆ’4=0 and the bottom row is a zero row, meaning that there are infinitely many solutions.

In most cases, it is impossible to immediately deduce whether a system of linear equations is consistent or inconsistent. It is therefore wisest to assume that a system is consistent and to begin finding the solution using the normal method of Gaussโ€“Jordan elimination to achieve an echelon form. Provided that we know how to recognize that a system is inconsistent whenever it becomes apparent, there is no disadvantage to proceeding as though the system is consistent. Within the broader scope of linear algebra, there are multiple ways of interpreting what it means for a system of equations to be inconsistent. However, one common way is to try and understand this physically. Earlier in the explainer, we had the system of linear equations ๐‘ฅ+๐‘ฆ=โˆ’3,2๐‘ฅ+2๐‘ฆ=โˆ’8, and we rearranged both of these straight line equations to get the two equations ๐‘ฆ=โˆ’3โˆ’๐‘ฅ,๐‘ฆ=โˆ’4โˆ’๐‘ฅ.

These two straight lines are parallel, meaning that they will never intersect. This is the framework that is used to understand what it means when a system of linear equations is inconsistent but has more variables or more equations. Generally speaking, we think of an inconsistent system as referring to straight lines, planes, hyperplanes, and so on that do not intersect each other. There is clearly nothing wrong with a situation where any two geometric objects do not meet each other, but it is necessary to have terminology to distinguish this case from the those where there is either a unique meeting point or infinitely many points.

Key Points

  • A system of equations is โ€œconsistentโ€ if there is at least one solution.
  • For a consistent system, there may be either a unique solution or infinitely many solutions (which are expressed in terms of independent parameters).
  • For an inconsistent system of linear equations, there are no viable solutions.
  • We recognize an inconsistent system when there is a pivot in the right-most entry of any echelon matrix corresponding to the original augmented coefficient matrix of the system.

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