Let's write 2 sets of parametric equations
that represent the solutions to the given linear equation.
And we're told to use the variable t as the parameter.
Because we have an equation and 2 variables,
the parametric equations that represent the solutions
are going to be in the form of x equals x of t
and y equals y of t.
We need to write both x and y as functions of t.
We're given a value of t.
The output of x and the output of y will satisfy
the given linear equation.
To do this,
it'll be helpful to solve the given equation for x or y.
In this case we'll do both
so that we can have 2 sets of parametric equations
that represent these solutions.
Let's first solve the given equation for x.
So beginning with 2x minus 4y equals zero,
to solve for x, we would first add 4y to both sides,
which gives us 2x equals 4y
and then divide both sides by 2.
Simplifying, we have x equals 2y.
Now that we have the equation solved for x,
to write the parametric equations,
the easiest thing to do is to let y equal t.
So if we let y equal t be one of the parametric equations,
if y is equal to t,
notice x is going to be equal to 2 times t or x equals 2t.
And now we have parametric equations
that represent these solutions to the given linear equation.
So in this form,
for any value of t,
the outputs of x and y will satisfy
the given linear equation.
To get the second set of parametric equations,
let's solve the same equation for y.
We could just divide x equals 2y by 2 on both sides
to solve for y,
but let's go ahead and start over
with the original equation.
Beginning with 2x minus 4y equals zero,
to solve for y,
we would subtract 2x on both sides,
which gives us negative 4y equals negative 2x,
and then divide both sides by negative 4.
Simplifying, we have y equals,
on the right negative 2 divided by negative 4 simplifies
to positive 1/2, giving us y equals 1/2 x.
Here, because we have the equation solved for y,
the easiest way to write the parametric equations
would be to let x equal t.
And notice, if x is equal to t,
then y is equal to 1/2 times t or y equals 1/2 t.
These parametric equations also represent the solutions
to the given linear equation.
Before we go,
let's take a look at a graph of the original equation,
as well as the parametric equations.
In blue we have the graph of the original linear equation
2x minus 4y equals zero.
Remember, in this form,
the lines do go in both directions forever.
And now let's take a look at the graph
of the parametric equations x equals 2t and y equals t.
Typically when graphing parametric equations,
we define an interval for t.
Here we have the closed interval for t
from negative 10 to positive 10.
So because we have a specific interval for t,
it's not going to graph the entire line.
If I turn this on,
we can see the parametric equations
and the resulting graph here in red.
Notice how it's only graphing a piece of the line,
which is actually a segment.
So as we change the interval for t,
it would change how much of the line is graphed.
If we did let t equal all real numbers,
then the red line would be exactly the same
as the blue line.
Also notice how all the points on the red segment
are on the blue line,
and therefore we know
our parametric equations are giving us solutions
to the original linear equation.
If we think of t as time,
we can think of the red segment being traced out
in a particular direction,
which is called the orientation of the graph,
given by the parametric equations.
So to show this,
I'm going to ahead and turn on the point 2a comma a
and then change the values of a,
which represent the parametric equations.
Notice when t is negative 10,
the x-value is negative 20, and the y-value is negative 10.
And as t increases from a negative 10 to a positive 10,
we can see the direction
in which the line segment is being traced out.
So you can think of this as a way to animate a point
along a particular path using parametric equations.
Again, each of these points are on the red line
as well as the blue line.
And therefore,
each ordered pair is a solution
to the original linear equation.
Now let's go ahead and turn these parametric equations off
and take a look at our second set,
which we're given by x equals t and y equals 1/2 t.
I'm going to go ahead and turn this on.
And the piece of the line that's graphed in black,
notice how it's not the same piece graphed
from the first set of parametric equations
over the same closed interval
from negative 10 to positive 10 for t.
And that's typical for different parametric equations
representing the same original equation.
But once again,
notice how all of the points on the black segment are
on the blue line,
and therefore they are all solutions
to the original linear equation.
I hope you found this helpful.