Question

Consider the parametric equations x=3t-5 and y=2t+3. How do you eliminate the parameter to find a Cartesian equation of the curve?

Answer 1

Hint: This question belongs to the topic of parametric functions from chapter calculus. In this question, first, we will find the value of t with respect to x. After that, we will find the value of t with respect to y. From there, we get 2 equations in which one is in terms of x and t and the other will be in the terms of y and t. From these two equations, we will equate them and get the equation in the terms of x and y. After that, we will get the Cartesian equation.

Complete step by step answer:
Let us solve this question.
In this question, we have given parametric equations which are x=3t-5 and y=2t+3. From these parametric equations, we have to remove the parameter t and we have to find a Cartesian equation.
So, from the equation x=3t-5, we will find the value of t with respect to x.
From the above equation, we can write
\[\Rightarrow \]x+5=3t
\[\Rightarrow \dfrac{x+5}{3}=t\]
From the above equation, we will put the value of t (as we have found above) in the equation y=2t+3 to get the Cartesian equation and to remove the parameter t.
So, we can write
y=2t+3
After putting the value of t in above equation, we get
\[\Rightarrow y=2\left( \dfrac{x+5}{3} \right)+3\]
We can write the above equation as
\[\Rightarrow y=\left( \dfrac{2\times x}{3} \right)+\dfrac{2\times 5}{3}+3\]
The above equation can also be written as
\[\Rightarrow y=\left( \dfrac{2}{3}x \right)+\dfrac{10}{3}+3=\dfrac{2}{3}x+\dfrac{10+9}{3}\]
Hence,
\[\Rightarrow y=\dfrac{2}{3}x+\dfrac{19}{3}\]

Note:
We should have a better knowledge of parametric equations and we should know how to find the Cartesian equation from the given parametric equations for solving this type of question easily. Don’t forget to remove the parameters. Parameters are very useful. Using parameters, we can find the equation of a circle, ellipse, and straight lines. So, the theory of parameters and parametric equations should be known.