Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far, we have seen several types of geometric relationships
that can exist when two linear equations are simultaneously graphed.
Simultaneously graphing systems of equations like this
is often a helpful tool in analyzing physical and mathematical relationships.
In particular, the points of intersection of two graphs have a special significance
since the intersection points represent common solutions to both equations.
To illustrate this concept, let's look at a race between a tortoise and a hare.
As the race begins, the hare, engrossed in a fascinating chapter in his Algebra book
decides to give the tortoise a head start
so that he can finish reading the chapter on "Solving Systems of Linear Equations".
As the race begins, the tortoise starts down the road at one-half mile per hour
a very brisk pace for a tortoise.
We can graph the tortoise's distance from the starting point as a function of time
letting the vertical axis represent distance
and the horizontal axis represent elapsed time.
We see that in the beginning when the elapsed time is zero
the tortoise's distance from the starting point is also zero.
After that, since he is travelling at a constant speed
his distance increases linearly as a function of time.
Since the tortoise's speed is one-half mile per hour
after four hours
he has travelled two miles.
Four hours after the race has begun, the hare finally finishes his Algebra chapter.
Even though the tortoise has obtained a significant head start
the confident hare decides to take it easy
since he knows, that even casually strolling at two miles per hour
he will eventually pass the tortoise.
So four hours after the race has begun
the hare is still zero miles from the starting line.
But from this point on, he travels at a constant speed of two miles per hour.
Since the hare travels at a faster rate than the tortoise
the hare's graph has a greater slope.
Looking at these two graphs, we see that throughout the race
the tortoise's distance increases linearly with time.
And after four hours of elapsed time
the hare's distance also increases linearly
until at some point between five and six hours after the race has begun
the hare and tortoise meet.
The point where the two graphs intersect
represents the instant at which the tortoise and the hare are at the same place
at the same time.
If we can create equations for these two graphs
there are mathematical techniques which we can use
to determine the coordinates of the point where the two graphs intersect.
Knowing these coordinates, we can then determine the exact time
and place
where the tortoise and the hare meet.
To create an equation for the tortoise's distance as a function of time
let's use the variable x to represent the time coordinate
and y to represent the distance coordinate.
We saw in the chapter on "Solving Motion Problems with Linear Equations"
that if we graph distance as a function of time for an object travelling at a constant speed
the slope of the graph will be equal to the speed of the object.
So the slope of this line is one-half.
Since we know that this graph intercepts the y-axis at the origin
we can use the slope-intercept form to write the equation for this line.
The value of the constant m in this equation is the slope of the line, one-half
and the value of b is the y-intercept, zero.
Eliminating the zero, we see that the equation for this line is "y equals one-half x".
Now that we have the equation for the tortoise's graph
we need to derive the equation for the hare.
Since the hare is travelling at a constant speed of two miles per hour
the slope of his graph is two.
We don't know the line's y-intercept
but we do know that the line passes through the point (4,0)
so we can use the point-slope form to write the equation for this line.
Filling in the values for the constants, we replace x-one in the equation with four
y-one with zero
and m with the line's slope, two.
We can then simplify the equation by eliminating the zero
and distribute the two to the terms in parentheses.
Multiplying two times four
to get eight
we see that the equation for this line is "y equals 2x minus eight".
We now have equations for the tortoise's and hare's distances as functions of time.
It is important to remember that although the lines described by these equations
extend infinitely in both directions
only the portion of the lines whose coordinates are positive
are meaningful in this particular example.
If we can now use these equations
to calculate the coordinates of the point where the two graphs intersect
we will be able to determine the exact time and distance
where the tortoise and the hare meet.
In the next lecture, we will introduce a technique called "substitution"
which will allow us to calculate the unique coordinates
which simultaneously solve both equations
and thus find the intersection point of the two graphs.