91Ó°ÊÓ

In mathematics, when we discuss the concept of a Cartesian equation, we're often dealing with equations that specify relationships between variables on the Cartesian coordinate system. The Cartesian coordinate system is the familiar grid-like structure where each point can be defined by an ordered pair (x, y).
Cartesian equations express how one variable is related to another without referencing a third variable or parameter.
In the context of our exercise, we have started with two parametric equations:

• \( x(t) = 2t - 1 \)
• \( y(t) = t^3 - 2 \)

The goal of eliminating the parameter \( t \) is to form a Cartesian equation, which helps in understanding the relation between \( x \) and \( y \) directly.

Eliminating the parameter from parametric equations is a technique used to transition from a parametric form to a simpler relational form.
This involves removing the third variable—in this case \( t \)—that links the two independent variables \( x \) and \( y \). By eliminating the parameter, we simplify the relationship and often make it easier to graph or analyze.
In our problem, we have two equations dependent on the parameter \( t \):

• \( x(t) = 2t - 1 \)
• \( y(t) = t^3 - 2 \)

The task is to express \( y \) in terms of \( x \), thereby eliminating \( t \) and creating a Cartesian equation that includes only \( x \) and \( y \). This approach provides greater insight into how changes in \( x \) affect \( y \) without the intermediary variable.

To eliminate a parameter effectively, the first step often involves solving for the parameter \( t \) in terms of one of the other variables—typically \( x \). This technique effectively means rearranging one of the parametric equations to reflect \( t = \) something involving \( x \).
For our parametric equation \( x(t) = 2t - 1 \), we need to isolate \( t \):

• Add 1 to both sides: \( x + 1 = 2t \)
• Divide everything by 2: \( t = \frac{x + 1}{2} \)

This gives us \( t \) in terms of \( x \), enabling us to substitute back into the other parametric equation. Solving for \( t \) is crucial because it forms the bridge that helps us transition from the parametric form to the Cartesian form.

After obtaining \( t \) in terms of \( x \), the substitution method allows us to replace \( t \) in the \( y(t) \) equation with our derived expression. This step is pivotal in eliminating \( t \) from the equations overall.
In our exercise, having determined that \( t = \frac{x + 1}{2} \), we proceed by substituting this value in the second equation: \( y(t) = t^3 - 2 \).
Replacing \( t \) results in:

• \( y = \left(\frac{x + 1}{2}\right)^3 - 2 \)

After substitution, simplify if necessary to achieve a neat and clear Cartesian equation:

• Expand the cube: \( \left(\frac{x + 1}{2}\right)^3 = \frac{(x+1)^3}{8} \)
• Combine results: \( y = \frac{(x+1)^3}{8} - 2 \)

This method of substitution retains the mathematical integrity while removing the complexity introduced by the parameter.