Hello. I'm Professor Von Schmohawk and welcome to Why U.
We have seen in the last several lectures that systems of two linear equations in two variables
can be visually represented by simultaneously graphing both equations.
We have also seen that the solutions to these systems fall into three categories
depending upon how the two lines in the graph intersect.
If the lines intersect at a single point
the system has a single unique solution which corresponds to that point.
A system composed of two distinct parallel lines
has no solutions, since distinct parallel lines don't intersect.
And a system composed of two identical lines
has an infinite number of solutions
since identical lines intersect at an infinite number of points.
These three types of systems are characterized by two attributes.
The systems can be either "consistent" or "inconsistent"
and the equations in the system can be either "dependent" or "independent".
Systems which have no solutions are said to be inconsistent
and systems with one or more solutions are called consistent
having either one solution
or an infinite number of solutions.
We can distinguish between these two types of consistent systems by their equations.
If the consistent system is composed of two distinct lines
the equations are independent
and if the consistent system is composed of two identical lines
the equations are dependent.
In the last lecture, we saw that the type of solutions we should expect
can be determined when the equations of the system are written in slope-intercept form.
In this lecture, we will see how to interpret the results we get when solving the system
to tell if the system has one solution
an infinite number of solutions
or no solutions.
In the lectures so far, we have seen how to solve systems of equations
which consist of two distinct intersecting lines.
Using either the substitution or elimination method
we end up with a pair of values for x and y
which is the single unique solution for the system.
The system is therefore a consistent system of independent equations.
The solution set of this system contains the single ordered pair (2,4).
Now let's see what happens when we attempt to solve a system of equations
composed of two distinct parallel lines.
Let's modify the equation used in the last example
by changing negative x in the bottom equation to positive x
which changes that equation's graph so that the two lines are now parallel.
Using the elimination method to solve this system
we first multiply the bottom equation by negative one
and then add the two equations to eliminate a variable.
But when we eliminate one variable, both variables are eliminated
and we get the equation "zero equals four".
A statement which can never be true
like this statement of equality between two non-equal numbers
is called a "contradiction".
Obtaining a contradiction when solving a system of equations
tells us that the system has no solutions and is therefore inconsistent.
Attempting to solve this system by the substitution method will also produce a contradiction.
Solving the bottom equation for x
we get "x equals two minus y"
which can then be substituted for x in the top equation
producing an equation with only the variable y.
The two y terms then cancel
leaving us with the contradiction "two equals six".
Once again, obtaining a contradiction indicates that this system has no solutions
and is therefore inconsistent.
Since the system has no solutions its solution set is the empty set.
Now let's see what happens when solving a system of equations composed of two identical lines.
This time, we'll make one equation a multiple of the other equation
so that the two graphs are identical.
Using the elimination method to solve this system
we can eliminate a variable by multiplying the bottom equation by negative one-half
and then add the two equations.
But when we eliminate one variable, every term is eliminated
producing the equation "zero equals zero".
Obtaining an equation which always must be true
like this statement of equality between two equal numbers
indicates that the equations of the system are dependent
and the system is therefore consistent
with an infinite number of solutions.
Solving this system by the substitution method
will also produce a statement of equality between two equal numbers.
We first solve the bottom equation for x by multiplying both sides by one-half
and then subtracting y from both sides.
We can then replace x in the top equation with the result
producing an equation with only the variable y.
The two y terms then cancel
leaving us with the statement "six equals six".
Once again, obtaining a statement of equality between two equal numbers
indicates that the equations are dependent.
The system therefore is consistent with an infinite number of solutions.
Since both equations have the same solution set
the solution set of this system is identical to the solution set of either equation.
Since the bottom equation is the simpler of the two
we will use it to define the solution set.
Using set builder notation
we can say that the solution set is the set of all ordered pairs (x,y)
such that "x plus y equals six".
In this lecture, we have seen how to interpret the results we get
when solving a system of two linear equations in two variables.
If solving the system produces a unique pair of values for x and y
we know that the system has one unique solution.
If solving the system produces a contradiction
the system has no solutions
and if solving the system produces a statement which is always true
the system has an infinite number of solutions.
As we have seen, any of these types of solutions
can occur when using a system of equations to solve real-world problems.
But any problem which can be solved using a system of equations
can also be solved by creating a single equation with a single variable.
In the next lecture, we will see what types of real-world problems
are easier to solve by using a system of equations.