91Ó°ÊÓ

A rectangular-coordinate equation converts parametric equations into a standard form involving only the variables \( x \) and \( y \). This form helps in understanding the nature of the curve or line on the Cartesian plane without needing to refer to a parameter like \( t \).

With parametric equations such as:

• \( x = 6t - 4 \)
• \( y = 3t \)

presented in the original exercise, the goal is to express \( x \) and \( y \) independently of \( t \).

After eliminating the parameter (which we will discuss next), we arrive at the rectangular equation \( x = 2y - 4 \). This equation describes a line, allowing one to easily identify its slope and intercept directly, thus providing a straightforward geometrical interpretation.

Eliminating the parameter is a key step in transforming parametric equations into a rectangular-coordinate equation. This involves resolving one of the equations for the parameter \( t \) and substituting it back into the other equation.

From the given parametric equations:

• \( x = 6t - 4 \)
• \( y = 3t \)

First, solve for \( t \) in terms of \( y \):
\[ t = \frac{y}{3} \]
Then, substitute \( t \) into the \( x \) equation:
\[ x = 6\left(\frac{y}{3}\right) - 4 \]
Simplifying gives you the rectangular equation:
\[ x = 2y - 4 \]
This process eliminates \( t \) and allows one to understand how \( x \) and \( y \) relate directly, forming a straight line when graphed.

Sketching on a coordinate plane involves plotting points obtained from parametric equations and connecting them to form a curve or line. This helps visualize the relationship between \( x \) and \( y \).

In the given exercise, begin by creating a table of values for different positive values of \( t \), since \( t \geq 0 \). For example:

• When \( t = 0 \), \( (x, y) = (-4, 0) \)
• When \( t = 1 \), \( (x, y) = (2, 3) \)
• When \( t = 2 \), \( (x, y) = (8, 6) \)

Plotting these points on a coordinate plane and connecting them results in a straight line. Because \( t \geq 0 \), the line begins at \(-4, 0\) and extends to the right. This visualization aids in understanding the linear relationship described by the rectangular equation \( x = 2y - 4 \).