91Ó°ÊÓ

In mathematics, eliminating parameters is a process where we remove the parameter, often denoted as \( t \), from a set of parametric equations. This allows us to express the relationship between the variables in a simpler form. Given the parametric equations:

• \( x(t) = 6 - 3t \)
• \( y(t) = 10 - t \)

We aim to find a relation solely between \( x \) and \( y \), without involving \( t \). This process generally involves solving one of the equations to express \( t \) in terms of one variable, and then substituting this expression into the other equation. This substitution eliminates the parameter and helps us find the Cartesian equation.

Cartesian equations describe the set of points (\(x, y\)) on a curve without the need for a parameter. They give a direct relationship between \(x\) and \(y\). After eliminating parameters, the parametric form \(x(t)\) and \(y(t)\) is converted into a cartesian equation. For example, once the parameter \(t\) from the equations \( x(t) = 6 - 3t \) and \( y(t) = 10 - t \) is eliminated, the resulting equation \( x = 3y - 24 \) describes the same curve using only \(x\) and \(y\). This form is often easier to analyse and visualize, as it directly associates \(x\) with \(y\) in a meaningful way across the entirety of the curve.

Solving equations for a variable is crucial in transforming parametric equations into a more digestible Cartesian form. Our first step in eliminating the parameter is to isolate \( t \) in terms of one of the variables. From the equation \( x = 6 - 3t \), rearranging gives us:- \( 3t = 6 - x \)- \( t = \frac{6 - x}{3} \)This equation provides a necessary substitution, removing \(t\) from our system. Isolating the parameter like this is a typical strategy in simplifying systems and forms the backbone of algebraic manipulation.

The substitution method is invaluable for eliminating the parameter and finding a Cartesian equation. Once we have \( t \) isolated, we swap it into the other parametric equation. For example, substituting \( t = \frac{6 - x}{3} \) into the second equation \( y = 10 - t \) results in:- \( y = 10 - \frac{6-x}{3} \)This operation allows us to express \( y \) directly in terms of \( x \), fully eliminating \( t \). After simplifying, the relation becomes evident, and further manipulation can clear fractions or arrange terms for simplicity.By substitution, we achieve a clear Cartesian equation \( x = 3y - 24 \), illustrating the elegant removal of the parameter \( t \) from the system.