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# finding rectangular equations from parametric equations

how do you find the rectangular equation for the plane curve defined by the parametric equations: x=t-3 and y=t^2+5?

### 2 Answers by Expert Tutors

Jim L. | Dr. Jim - Harvard graduate tutor - perfect SAT score!Dr. Jim - Harvard graduate tutor - perfe...
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Hi Morgan,

Converting parametric equation to a cartesian equation or rectangular form involves solving for t in terms of x and then plugging this into the y equation.  Therefore,

t=x+3

y = (x+3)^2 + 5

y = x^2+6x + 9 + 5

y= x^2 + 6x + 14

Hope that helps.  Jim

Isaak B. | Good (H.S. or College Math, Physics, Chem, EE Engineering) Cheap TutorGood (H.S. or College Math, Physics, Che...
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Yes, Jim is entirely correct.

I would only augment his answer by pointing out that in general you need to manipulate the parametric equations to eliminate the parametric variable.

It made the most sense to eliminate t by solving the first equation for t, in this particular problem, because that equation was linear so each value of x only corresponded to one value of t.  Things would have gotten a bit messy in this case if you had tried to solve the second equation for t, in this problem, but in general you eliminate the non rectangular-grid parameter by whatever means possible if you want the expression converted to its rectilinear coordinates.

Which equation should be solved for the parametric variable depends on the problem -- whichever equation can be most easily solved for that parametric variable is typically the best choice.  For instance had the problem been y = t -3, and x = t^2 + 5, I hope you see that solving for t in terms of y would make more sense, for exactly the same reasons already discussed.

So you solve for t in terms of x if that's easier, or solve for t in terms of y if that's easier, then put the result into the other equation. In this problem, the former was easier.