1.1: Systems of Linear Equations

Learning Objectives

• Define linear equations and systems of linear equations.
• Classify systems of linear equations based on the existence and uniqueness of solutions.
• Interpret geometrically two-variable systems as intersecting, parallel, or coinciding lines.
• Extend geometric understanding to three-variable systems, including intersecting, non-intersecting, or coinciding planes.
• Recognize the increasing complexity of systems as the number of variables grows and the need for linear algebra techniques to analyze them.

We will start with a definition of a linear equation. You may have encountered them before, but here is a general definition that we will use.

Definition \(\PageIndex{1}\): Linear

A linear equation in the variables \(x_1,x_2,\dots ,x_n\) is an equation that can be written in the form

\[a_1x _1 + a_2x_2+ \dots +a_nx_n=b\]

where \(a_1 , a_2, \dots , a_n\) are called coefficients and b is called constant all are real or complex numbers.

Example \(\PageIndex{1}\)

Whic of the following equations is a linear equation

\[3x + 4y = 2z \]

\[x_1x_2 + 3x_4 = x_3 \]

Solution

The equation \(3x + 4y = 2z \) is linear since it's satisfy the definion bove. However \(x_1x_2 + 3x_4 = x_3 \) since the two variables \(x_1 \) and (x_2 \) are multiplied together.

We will usually move the unknowns to the left side of the equation, and move the constants to the right.

Definition \(\PageIndex{1}\): System of Linear Equations

A system of linear equations is a list of equations, \[\begin{array}{c} a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}=b_{1} \\ a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}=b_{2} \\ \vdots \\ a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}=b_{m} \end{array}\nonumber\] where \(a_{ij}\) and \(b_{j}\) are real numbers. The above is a system of \(m\) equations in the \(n\) variables, \(x_{1},x_{2}\cdots ,x_{n}\). Written more simply in terms of summation notation, the above can be written in the form \[\sum_{j=1}^{n}a_{ij}x_{j}=b_{i}, \text{ }i=1,2,3,\cdots ,m\nonumber\]

The relative size of \(m\) and \(n\) is not important here. Notice that we have allowed \(a_{ij}\) and \(b_{j}\) to be any real number. We can also call these numbers scalars . We will use this term throughout the text, so keep in mind that the term scalar just means that we are working with real numbers.

Example \(\PageIndex{1}\)

Consider the system of linear equations, \[\left\{\begin{array}{rrrrc} x &+& 2y &=& 3 \\ y &+& 2z &=& -3.\end{array}\right.\nonumber\]

how many variables are involved in the system

Solution

The system consists of two equations and three variables, namely x, y, and z, since each variable appears at least once in the system.

Definition \(\PageIndex{4}\): \(\mathbb{R}\)

Let \(n\) be a positive whole number. We define

\[ \mathbb{R}^n = \text{all ordered \(n\)-tuples of real numbers }(x_1,x_2,x_3,\ldots,x_n). \nonumber \]

An \(n\)-tuple of real numbers is called a point of \(\mathbb{R}^n\).

In other words, \(\mathbb{R}^n\) is just the set of all (ordered) lists of \(n\) real numbers. We will draw pictures of \(\mathbb{R}^n\) in a moment, but keep in mind that this is the definition. For example, \((0, \frac 32, -\pi)\) and \((1,-2,3)\) are points of \(\mathbb{R}^3\). Also the point \((0,0,0,0,0)\) is in \(\mathbb{R}^5\) and it has a spesific name the origin of \(\mathbb{R}^5\).

Definition \(\PageIndex{2}\): Solution sets

• A solution of a system of equations is the point \((x_1,x_2, \ldots, x_n)\) that make all of the equations true simultaneously.
• The solution set of a system of equations is the collection of all solutions.
• Solving the system means finding all of the solutions.

A system of linear equations can have one unique solution, infinitely many solutions, or no solution at all. Below are the terminologies used to describe the two main categories of solutions for any given system of linear equations.

Definition \(\PageIndex{3}\): Inconsistent and Consistent

A system of equations is called inconsistent if it has no solutions. It is called consistent otherwise.

Example \(\PageIndex{1}\)

Determine whether the system is consistent or inconsistent:

\[
\begin{cases}
2x - y = -1 \\
2x - y = 3
\end{cases}
\]

\textbf{Solution:}

We will use elimination to solve. First multiply the send equation by negative and then add the two equations
\[
\begin{array}{rcl}
2x - y & = & -1 \\
-(2x - y & = & 3) \\ \hline
0 & = & -4
\end{array}
\]

\textbf{Conclusion:} Since we arrived at a false statement, there is no point of intersection, and the system is inconsistent. Graphically, these lines are parallel. You can plot the two lines to confirm that they never intersect.

Example \(\PageIndex{1}\)

Determine whether the following system of equations is consistent or inconsistent. Solve the system using back-substitution.

\[
\begin{cases}
x + 2y + 3z = 9 \\
y + z = 3 \\
z = 1
\end{cases}
\]

Solution

In this system, we already know the value of \(z\), and the second equation involves only \(y\) and \(z\). The fastest way to solve is:

1. Substitute \(z = 1\) into the second equation to solve for \(y\):

\[
y + 1 = 3 \implies y = 2
\]

2. Substitute \(y = 2\) and \(z = 1\) into the first equation to solve for \(x\):

\[
x + 2(2) + 3(1) = 9 \implies x + 4 + 3 = 9 \implies x = 2
\]

Thus, we have the unique point in \(\mathbb{R}^3\) as the solution:

\[
(x, y, z) = (2, 2, 1)
\]

Thus, the system is consistent.

Systems of Equations, Geometry

Outcomes

1. Relate the types of solution sets of a system of two (three) variables to the intersections of lines in a plane (the intersection of planes in three space)

As you may remember, linear equations like \(2x+3y=6\) can be graphed as straight lines in the coordinate plane. We say that this equation is in two variables, in this case \(x\) and \(y\). Suppose you have two such equations, each of which can be graphed as a straight line, and consider the resulting graph of two lines. What would it mean if there exists a point of intersection between the two lines? This point, which lies on both graphs, gives \(x\) and \(y\) values for which both equations are true. In other words, this point gives the ordered pair (\(x,y\)) that satisfy both equations. If the point \(\left( x, y \right)\) is a point of intersection, we say that \(\left( x, y \right)\) is a solution to the two equations. In linear algebra, we often are concerned with finding the solution(s) to a system of equations, if such solutions exist. First, we consider graphical representations of solutions and later we will consider the algebraic methods for finding solutions.

When looking for the intersection of lines on a graph, several situations may occur. The following graphs shows the possible cases for two equations (top three graphs) and three equations (bottom two graphs), each involving two variables.

 

Figure \(\PageIndex{1}\)

This illustration shows the possible solutions of a linear system. In the first case, graphs labeled “one solution” show lines intersecting at a single point, representing a unique solution to the system. In the second case, graphs labeled “no solution” show lines that do not intersect—either because they are parallel or because not all lines meet at the same point—indicating that there is no solution. In the third case, graphs labeled “infinitely many solutions” show lines that coincide, meaning they are the same line; here, every point on the line satisfies both equations, resulting in infinitely many solutions.

Example \(\PageIndex{1}\): A Graphical Solution

Use a graph to find the solution to the following system of equations \[\begin{array}{c} x+y=3 \\ y-x=5 \end{array}\nonumber \]

Solution

Through graphing the above equations and identifying the point of intersection, we can find the solution(s). Remember that we must have either one solution, infinitely many, or no solutions at all. The following graph shows the two equations, as well as the intersection. Remember, the point of intersection represents the solution of the two equations, or the \(\left( x,y\right)\) which satisfy both equations. In this case, there is one point of intersection at \(\left( -1, 4 \right)\) which means we have one unique solution, \(x = -1, y = 4\).

Figure \(\PageIndex{2}\)

In the above example, we investigated the intersection point of two equations in two variables, \(x\) and \(y\). Now we will consider the graphical solutions of three equations in two variables.

Consider the first picture above. While all three lines intersect with one another, there is no common point of intersection where all three lines meet at one point. Hence, there is no solution to the three equations. Remember, a solution is a point \(\left( x, y \right)\) which satisfies all three equations. In the case of the second picture, the lines intersect at a common point. This means that there is one solution to the three equations whose graphs are the given lines. You should take a moment now to draw the graph of a system which results in three parallel lines. Next, try the graph of three identical lines. Which type of solution is represented in each of these graphs?

We have now considered the graphical solutions of systems of two equations in two variables, as well as three equations in two variables. However, there is no reason to limit our investigation to equations in two variables. We will now consider equations in three variables.

You may recall that equations in three variables, such as \(2x+4y-5z=8\), form a plane. Above, we were looking for intersections of lines in order to identify any possible solutions. When graphically solving systems of equations in three variables, we look for intersections of planes. These points of intersection give the \(\left( x, y, z \right)\) that satisfy all the equations in the system. What types of solutions are possible when working with three variables? Consider the following picture involving two planes, which are given by two equations in three variables.

Figure \(\PageIndex{4}\)

Notice how these two planes intersect in a line. This means that the points \(\left( x,y,z\right)\) on this line satisfy both equations in the system. Since the line contains infinitely many points, this system has infinitely many solutions.

It could also happen that the two planes fail to intersect. However, is it possible to have two planes intersect at a single point? Take a moment to attempt drawing this situation, and convince yourself that it is not possible! This means that when we have only two equations in three variables, there is no way to have a unique solution! Hence, the types of solutions possible for two equations in three variables are no solution or infinitely many solutions.

Now imagine adding a third plane. In other words, consider three equations in three variables. What types of solutions are now possible? Consider the following diagram.

Figure \(\PageIndex{5}\)

In this diagram, there is no point which lies in all three planes. There is no intersection between all planes so there is no solution. The picture illustrates the situation in which the line of intersection of the new plane with one of the original planes forms a line parallel to the line of intersection of the first two planes. However, in three dimensions, it is possible for two lines to fail to intersect even though they are not parallel. Such lines are called skew lines.

Recall that when working with two equations in three variables, it was not possible to have a unique solution. Is it possible when considering three equations in three variables? In fact, it is possible, and we demonstrate this situation in the following picture.

Figure \(\PageIndex{6}\)

In this case, the three planes have a single point of intersection. Can you think of other types of solutions possible? Another is that the three planes could intersect in a line, resulting in infinitely many solutions, as in the following diagram.

Figure \(\PageIndex{7}\)

We have now seen how three equations in three variables can have no solution, a unique solution, or intersect in a line resulting in infinitely many solutions. It is also possible that the three equations graph the same plane, which also leads to infinitely many solutions.

You can see that when working with equations in three variables, there are many more ways to achieve the different types of solutions than when working with two variables. It may prove enlightening to spend time imagining (and drawing) many possible scenarios, and you should take some time to try a few.

You should also take some time to imagine (and draw) graphs of systems in more than three variables. Equations like \(x+y-2z+4w=8\) with more than three variables are often called hyper-planes. You may soon realize that it is tricky to draw the graphs of hyper-planes! Through the tools of linear algebra, we can algebraically examine these types of systems which are difficult to graph. In the following section, we will consider these algebraic tools.

Exercise \(\PageIndex{1}\)

Graphically, find the point \((x_1,y_1)\) which lies on both lines, \(x+3y=1\) and \(4x-y=3\). That is, graph each line and see where they intersect.

Exercise \(\PageIndex{2}\)

Graphically, find the point of intersection of the two lines, \(3x+y=3\) and \(x+2y=1\). That is, graph each line and see where they intersect

Exercise \(\PageIndex{3}\)

You have a system of \(k\) equations in two variables, \(k ≥ 2\). Explain the geometric significance of

1. No solution.
2. A unique solution.
3. An infinite number of solutions.