Sketch the curve represented by the parametric equations...

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by climinating the parameter.
$$x=2 t-3, \quad y=3 t+1$$

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of one or more parameters. Instead of directly relating x and y, the equations x = 2t - 3 and y = 3t + 1 define both variables in terms of a third variable t, which can often simplify the representation of curves, especially when describing motion or more complex trajectories.

Elimination of the Parameter

Eliminating the parameter involves solving one of the parametrically defined equations for the parameter and substituting that expression into the other equation. This process converts the parametric equations into a single equation in x and y, known as the rectangular or Cartesian form of the equation. For linear equations like these, the parameter can be easily isolated and removed.

Rectangular Equations

A rectangular equation expresses the relationship between x and y directly, without the use of an extra parameter. Once the parameter t is eliminated from the given equations, the resulting equation describes a straight line in the Cartesian plane. This form is useful for analyzing the overall structure and properties of the curve.

Curve Sketching and Orientation

Sketching the curve involves plotting the relationship described by the rectangular equation in the x-y plane. Additionally, understanding the orientation of the curve means determining the direction in which the curve is traced as the parameter increases. In this context, the orientation is determined by the parametric definitions and directional rates of change in x and y with respect to t.

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