Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last several lectures, we studied systems of two linear equations in two variables
and saw how these systems could be represented graphically as lines in the Cartesian plane.
Systems of equations can consist of more than two equations
and the equations in the system can contain more than two variables.
In this lecture, we will introduce systems of three linear equations in three variables
and see how they can be visualized in 3-dimensional Cartesian space.
The standard form of a linear equation in x and y is written as "Ax + By = C"
where A, B, and C are constants
and at least one of the coefficients, A or B must be non-zero.
The term "linear" reflects the fact that the graph of an equation of this form
will be a straight line in the xy-plane.
Just as a standard form linear equation in two variables x and y
is written as "Ax + By = C"
a standard form linear equation in three variables x, y, and z
is written as "Ax + By + Cz = D"
where at least one of the coefficients A, B, or C must be non-zero.
Since the equation contains three variables
it must be graphed using a 3-dimensional coordinate system
as we saw in the lecture "Cartesian Coordinates in Three Dimensions".
So what does the graph of a linear equation in three variables look like?
Let's start by looking at some simple examples in two variables
and then extend the examples to three variables.
When a linear equation is written in standard form
setting the coefficient of any variable to zero
causes the term containing that variable to disappear.
Even though that variable is no longer included in the equation
it is understood that the variable still exists and is free to take on any value.
This is why in a standard form linear equation, A and B cannot both be zero
since this would eliminate both variables from the equation.
If, in fact, the constant C was zero, then x and y could take on any value
and the statement would still be true.
The solution set would therefore contain every possible combination of x and y.
And if the constant C was not zero
then no values of x and y would make the statement true
and the solution set would therefore be the empty set.
Since in either case, the graph of the solution set would not be a line
the equation "0 = C" cannot be a linear equation.
So let's create a simple linear equation in x and y
by setting A to zero, and B to one.
This eliminates x from the equation.
If we set the constant C to three
we get the linear equation "y = 3".
Since the values of x are no longer constrained by the equation
x is free to take on any value.
The graph of this equation therefore consists of every point whose y-coordinate is three
and x-coordinate is any real number.
The set of all these points is a horizontal line
three units above the origin.
Likewise, setting B to zero, and A to one
eliminates y from the equation.
If once again, we set C to three
we get the linear equation "x = 3".
Now since y is free to take on any value
the graph of this equation consists of every point whose x-coordinate is three
and y-coordinate is any real number.
The graph of this equation is therefore a vertical line
three units to the right of the origin.
Now let's try the same thing for a linear equation in three variables
setting the coefficients of all but one of the variables to zero.
Let's start by setting A and B to zero, and C to one.
This eliminates all the variables but z.
If we set the value of D to three
we get the linear equation "z = 3".
Since both x and y are free to take on any value
the graph of this equation consists of every point in 3-dimensional space
whose z-coordinate is three
and x and y-coordinates are any real number.
The graph of this equation is therefore a horizontal plane
three units above the origin.
If we set A and C to zero, and B to one
all variables but y are eliminated from the equation.
And setting D to three
we get the equation "y = 3".
Now since x and z are free to take on any value
the graph of this equation consists of every point whose y-coordinate is three.
The graph of this equation is therefore a vertical plane
three units to the right of the origin.
Likewise, if we set A, B, C and D to one, zero, zero, and three
all variables but x are eliminated
giving us a graph which consists of all points with an x-coordinate of three.
The graph of this equation is a vertical plane
three units in front of the origin.
Now what do you suppose would happen
if we set the coefficient of only one of the three variables to zero?
For example, let's set the z coefficient to zero
and the x and y coefficients to one.
This eliminates z from the equation
but leaves the variables x and y.
Now, if we set the value of D to three
we get the linear equation "x + y = 3".
Let's ignore z for the moment and plot this equation on the xy-plane.
This gives us a line which intersects the x and y axes.
Every point on this line is a solution to the equation "x + y = 3"
where the z-coordinate of every point is zero.
But there are more solutions than this.
All points on any line where x + y = 3 are solutions to this equation
regardless of the value of z.
There are an infinite number of lines like this
one line for every possible value of z.
Since z can be any real number, all these lines together create a plane.
Each point in this plane corresponds to a solution of the linear equation "x + y = 3".
The graph of a linear equation in three variables
will always be a plane when graphed in Cartesian coordinates
assuming that constants A, B, and C are not all zero.
Any plane which corresponds to the solutions extends infinitely in all directions.
Depending upon the values of A, B, C, and D
the plane may lie in any position and orientation.
We have seen that systems of two linear equations in two variables
can be represented graphically as lines in the Cartesian plane
and the solutions of these systems correspond to the points where the lines intersect.
In systems of three linear equations in three variables
the equations are represented as planes in Cartesian space.
Each plane may be oriented at any angle
and positioned anywhere in space relative to the other planes.
Since each plane graphically represents the set of solutions to one of the three equations
the points where all three planes simultaneously intersect
correspond to solutions which simultaneously satisfy all three equations.
These points therefore correspond to the solutions of the system.
In the next lecture, we will look at different ways
in which three planes can intersect in 3-dimensional space
and thus different types of solutions which can exist
for a system of three linear equations in three variables.