Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lectures, we saw how a race between a tortoise and a hare
could be mathematically represented by a system of two linear equations.
Graphing their distances from the starting point as functions of time
the tortoise starts first, travelling at a constant speed of one-half mile per hour.
After giving the tortoise a four-hour head start, the hare begins the race
travelling at a constant speed of two miles per hour.
Since the hare's speed is greater than that of the tortoise
the slope of the hare's graph is greater, causing the two graphs to intersect.
This intersection represents the point at which the tortoise and hare meet.
We showed how this system of linear equations could be solved
either by using the "substitution" method
or the "elimination" method.
Solving this system of equations, gives us the coordinates of the intersection point
and thus the distance and time when the tortoise and hare are both at the same place
at the same time.
However, not all races where the tortoise and hare both travel at constant speeds
would result in their meeting at one point in time.
For example, let's say that instead of travelling at one-half mile per hour
the tortoise decides to use roller skates
which allow him to travel at the same speed as the hare.
Since the two runners' speeds are the same
their graphs have the same slope
and the two lines are parallel.
Since the two graphs never intersect, the tortoise and hare never meet.
This system of linear equations therefore has no solution.
Yet another possibility is that both the tortoise and hare start the race at the same time
and travel at the same speed.
Their graphs therefore coincide
reflecting the fact that the tortoise and hare both have identical distance values
at every point in time during the race.
Since the two graphs intersect at every point
there are an infinite number of solutions to this system of equations.
So there are three possible types of solutions we may encounter
when solving systems of two linear equations in two variables.
The first possibility is that the two graphs cross.
Their single intersection point reflects the fact that only a single solution exists
which simultaneously satisfies both equations.
The second possibility is that the two equations describe distinct but parallel lines.
Since these graphs have no point of intersection
no solution exists which can satisfy both equations.
The solution set of this system is therefore the empty set.
The third possibility is that the two lines described by the equations are the same.
In this case, the two graphs intersect at every point
and the solution set contains an infinite number of solutions.
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".
So what exactly do these terms mean?
Any system of equations for which no solution exists is said to be inconsistent.
And any system with one or more solutions is said to be consistent.
The consistent systems we have seen may have a single solution
when the equations describe different lines
or an infinite number of solutions
when the equations both describe the same line.
Equations which describe different lines are said to be independent
and equations which describe the same line are called dependent.
So consistent systems with independent equations have one solution
consistent systems with dependent equations have an infinite number of solutions
and inconsistent systems have no solutions.
We can quickly determine which type of solutions we should expect
by writing the equations in slope-intercept form.
Let's say we are given a system of two linear equations in two variables
written in standard form.
If we rewrite the equations in slope-intercept form y = mx + b
by examining the constants m and b
which correspond to a graph's slope and y-intercept
we can tell whether the system has one solution, no solutions, or an infinite number of solutions.
If m is the same in both equations
and b is different
then the lines have the same slope but different y-intercepts
so the lines are parallel.
Since distinct parallel lines don't intersect, the system has no solutions
and is therefore inconsistent.
On the other hand, if the equations have different values for m
then the slope is different for each equation.
Regardless of the value of b
the lines cannot be parallel and must intersect somewhere at a single point.
The system therefore has one solution and is consistent
and since each equation describes a different line
the equations are independent.
If the equations of the system both describe the same line
then they must have the same slope and y-intercept
so when written in slope-intercept form, the equations will be identical.
Therefore the system has an infinite number of solutions.
Since the system has one or more solutions, it is consistent
and since both equations describe the same line, the equations are dependent.
In previous lectures, we saw how to use substitution or elimination
to solve systems of two linear equations in two variables
whose solution set consists of a single solution.
In the next lecture, we will see how to interpret the results we get
when solving systems with an infinite number of solutions
or no solutions.