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Parametric equations are a way to express mathematical equations where the position of a point is determined by one or more parameters. In simpler terms, instead of describing a curve with a single equation in terms of two variables, we use separate equations for each variable, all defined with respect to another common variable, called the parameter.

For example, in the given exercise, the parametric equations are \( x(t) = 4 - t \) and \( y(t) = 3t + 2 \), where \( t \) is the parameter. This means as \( t \) changes, both \( x \) and \( y \) change, describing a point on the plane. This method is especially useful for describing curves that may have complex and multiple values implicitly in one coordinate system.

Using parametric equations allows for precise control over each variable independently, which can then be converted into a single Cartesian equation for a unified representation.

A Cartesian equation represents a relationship between two coordinates (commonly known as \( x \) and \( y \)) on a plane without using a parameter. These equations are what we typically think of when imagining linear graphs, circles, or parabolas.

In the exercise, once we convert the parametric equations into a Cartesian form, we end up with the equation \( y = 14 - 3x \). Here, \( y \) is directly expressed in terms of \( x \). This allows us to easily graph or analyze the curve described by these coordinates on a standard \( xy \)-plane.

The main aim of transforming parametric equations into this form is to simplify the graphical representation to something that is more familiar and intuitive.

The substitution method is a valuable technique for transforming parametric equations into a Cartesian equation. This involves expressing one of the parameters from the parametric equations in terms of another variable and then substituting back to eliminate the parameter.

In our exercise, we began with \( x(t) = 4 - t \) and expressed \( t \) in terms of \( x \), giving us \( t = 4 - x \). This expression of \( t \) was then substituted into the second equation \( y(t) = 3t + 2 \) to eliminate the parameter \( t \).

This step is crucial because it connects the independent parametric equations into one comprehensive Cartesian form. Having a unified equation simplifies analysis and makes it easier to graphically interpret the solution.

Simplifying equations is the final yet critical step in transforming parametric equations to Cartesian form. It involves performing operations to make the equation more straightforward and easier to read or analyze.

Once we substituted \( t = 4 - x \) into the equation \( y = 3(4 - x) + 2 \), we then rearranged and simplified it by performing basic algebraic operations:

• Distributed the 3 into the terms inside the parenthesis.
• Combined like terms, which resulted in \( y = 12 - 3x + 2 \).
• Simplified further to achieve \( y = 14 - 3x \).

This yields a clean Cartesian equation free of unnecessary complexity. Simplifying makes analysis more intuitive and provides a clear mathematical representation of the relationship between the variables.