Eliminating time (t) between two equations

[ohms law]

I'm working through several example problems, and one thing that has come up a couple of times is where the person who solved these problems says "Eliminating t between equations (1)
and (2) yields", or something similar. He's deriving an equation based on two other equations obviously, but I don't understand how or why (well, I sort of understand why, but...)

So, in this instance, equation (1) is:
$x=V_{0x}t=(V_{0}Cos\Theta_{0})t$

Equation (2) is:
$y=y_{0}+V_{0y}t+{1/2}a_{y}t^{2}$

So, smashing them together (however you do that) and "eliminating t" (whatever that means, beyond the obvious), yields:
$y=y_{0}+(Tan\Theta_{0})x+({a_{y}/2v_{0}^{2}Cos^{2}\Theta_{0}})$
I mean... I basically have that memorized now, but... how?

 

[SHISHKABOB]

it looks like he solved for t in the first equation and then substituted that in for t in equation 2

 

[azizlwl]

Notes on parametric equations.
http://tutorial.math.lamar.edu/Classes/CalcII/ParametricEqn.aspx

 

[ohms law]

ooooh, parametric equations... haven't started those yet (in my calc class).
Thanks azizlwl!

That's actually a good insight too, shishkabob.
So, thanks to both of you.

(actually, after skimming through that parametric equations tutorial, I think that we've started on this material... some of it, anyway. humm...)

 

[Nickie]

Eliminating time between two equations means finding a way to express one variable (in this case, time) in terms of the other variables in the equations. This is often done to simplify the equations and make them easier to solve or manipulate.

In this example, the goal is to eliminate t from equations (1) and (2) to get an equation that only involves x, y, and the other constants (V0x, V0, θ0, y0, V0y, and ay).

To eliminate t, we can rearrange equation (1) to solve for t:
t=x/(V0cosθ0)

Then we can substitute this value for t into equation (2):
y=y0+(V0sinθ0)x/(V0cosθ0)+1/2ay(x/(V0cosθ0))^2

Simplifying this equation gives us:
y=y0+tanθ0x+1/2ayx^2/V0^2cos^2θ0

Which is the same as the equation you have memorized, except for a minor difference in notation.

In summary, by eliminating time between equations (1) and (2), we are able to express y in terms of x and the other constants, simplifying the problem and making it easier to solve. This technique is commonly used in physics and other scientific fields to manipulate equations and solve problems.