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.