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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 a system of three equations in two variables. Again, these equations can be graphed as

straight lines in the plane, so that the resulting graph contains three straight lines. Recall the three possible

types of solutions; no solution, one solution, and infinitely many solutions. There are now more complex

ways of achieving these situations, due to the presence of the third line. For example, you can imagine

the case of three intersecting lines having no common point of intersection. Perhaps you can also imagine

three intersecting lines which do intersect at a single point. These two situations are illustrated below.

x

y

No Solution

x

y

One Solution

1.1. Systems of Equations, Geometry 5

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 (x,y) 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 (x,y,z) 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.

Notice how these two planes intersect in a line. This means that the points (x,y,z) 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.

✠ New Plane

In this diagram, there is no point which lies in all three planes. There is no intersection between all

6 Systems of Equations

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.

✠

New Plane

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.

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.

1.2. Systems Of Equations, Algebraic Procedures 7

Exercises

Exercise 1.1.1 Graphically, find the point (x1,y1) which lies on both lines, x+ 3y = 1 and 4x− y = 3. That is, graph each line and see where they intersect.

Exercise 1.1.2 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 1.1.3 You have a system of k equations in two variables, k ≥ 2. Explain the geometric signifi- cance of

(a) No solution.

(b) A unique solution.

(c) An infinite number of solutions.

1.2 Systems Of Equations, Algebraic Procedures

Outcomes

A. Use elementary operations to find the solution to a linear system of equations.

B. Find the row-echelon form and reduced row-echelon form of a matrix.

C. Determine whether a system of linear equations has no solution, a unique solution or an

infinite number of solutions from its row-echelon form.

D. Solve a system of equations using Gaussian Elimination and Gauss-Jordan Elimination.

E. Model a physical system with linear equations and then solve.

We have taken an in depth look at graphical representations of systems of equations, as well as how to

find possible solutions graphically. Our attention now turns to working with systems algebraically.

8 Systems of Equations

Definition 1.2: System of Linear Equations

A system of linear equations is a list of equations,

a11x1 +a12x2 + · · ·+a1nxn = b1 a21x1 +a22x2 + · · ·+a2nxn = b2

...

am1x1 +am2x2 + · · ·+amnxn = bm

where ai j and b j are real numbers. The above is a system of m equations in the n variables,

x1,x2 · · · ,xn. Written more simply in terms of summation notation, the above can be written in the form

n

∑ j=1

ai jx j = bi, i = 1,2,3, · · · ,m

The relative size of m and n is not important here. Notice that we have allowed ai j 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.

Now, suppose we have a system where bi = 0 for all i. In other words every equation equals 0. This is a special type of system.

Definition 1.3: Homogeneous System of Equations

A system of equations is called homogeneous if each equation in the system is equal to 0. A

homogeneous system has the form