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Parametric equations describe a curve or path by specifying coordinates \(x\) and \(y\) as functions of a parameter, typically \(t\). Converting them into a Cartesian equation involves eliminating the parameter to express \(y\) explicitly in terms of \(x\). To achieve this, separate and solve one of the original equations for \(t\), and then substitute this expression into the other equation. For example, given the parametric equations \(x = 3t + 1\) and \(y = t - 4\), solving the first equation gives \(t = \frac{x - 1}{3}\). Substituting this into the second equation transforms the parametric form into a single equation in terms of \(x\) and \(y\).The result is \(y = \frac{x - 13}{3}\), an equation in Cartesian form. This form gives a clear picture of the curve, allowing us to analyze relationships between \(x\) and \(y\) without the need for the parameter \(t\). Therefore, Cartesian equations are essential for simplifying complex motion into understandable mathematical relationships.

The equation of a curve provides a way to represent a curve in a single, cohesive formula. This formula defines how the \(x\) coordinate relates to the \(y\) coordinate along a specific path. In the context of this exercise, converting parametric equations to a curve equation means synthesizing two separate relationships into one. By substituting the expression for \(t\) into the equation of \(y\), a continuous curve equation \(3y = x - 13\) is created. This means that for any point \( (x, y) \) on the curve, this equation must hold true.Understanding these curve equations helps visualize the path or trajectory of the particle. So whether it be a line, parabola, or more complex shape, converting to an equation of a curve makes the interpretation and analysis more straightforward.

In particle motion studies, parametric equations are used to describe the movement of an object in terms of time. Each component, \(x(t)\) and \(y(t)\), illustrates how each coordinate changes as time progresses. For example, with \(x = 3t + 1\) and \(y = t - 4\), we see that the particle moves along a straight line in the plane. Here, \(t\) represents time, suggesting how the position of a particle evolves. Breaking down these equations demonstrates the individual motion along each axis:

• The \(x\)-coordinate shifts at a constant rate proportional to \(t\).
• The \(y\)-coordinate also changes linearly but with a different rate.

By eliminating \(t\) and deriving a Cartesian form, we consolidate this into a simpler representation of the path, \(x = 3y + 13\). This consolidation is critical in studying motion as it highlights the overall trajectory of the particle without being obscured by the parameter, allowing for easier analysis of the path taken by the particle over time.