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Find a Cartesian equation for the parametric equation x=t, \quad y = 9t + 3

Question:

Find a Cartesian equation for the parametric equation

{eq}x=t, \quad y = 9t + 3 {/eq}

Cartesian Equation of a Curve:

We are given two parametric equations {eq}x= f(t) {/eq} and {eq}y= g(t) {/eq}. We need to compute the Cartesian equation of the given curves.

To solve, we'll write t in terms of x and y and equate the values of the parameter and apply algebraic rules to simplify the answer.

Answer and Explanation:

We are given:

{eq}x=t, \quad y = 9t + 3 {/eq}


The first equation can be written as {eq}t=x {/eq}


Isolate t from the second equation: {eq}y = 9t + 3 \Rightarrow 9t =y -3 \Rightarrow t =\dfrac{y-3}{9} \\ {/eq}

Now equating both values of {eq}t: {/eq}

{eq}\Rightarrow x = \dfrac{y-3}{9} {/eq}

{eq}\Rightarrow 9x = y-3 {/eq}

{eq}\Rightarrow y= 9x+3 {/eq}


Therefore, the Cartesian equation of the curve is {eq}{\boxed{ y= 9x+3.}} {/eq}


Learn more about this topic:

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Parametric Equations in Applied Contexts

from Precalculus: High School

Chapter 24 / Lesson 6
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