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Eliminating parameters is a useful technique when dealing with parametric equations. It allows you to simplify a set of parametric equations into a single equation in terms of just one variable. This process makes it easier to analyze and graph the curve.

In the exercise given, we have two parametric equations:

• \( x = t - 3 \)
• \( y = 3t - 7 \)

The first step to eliminate the parameter "\( t \)" is to express it in terms of one of the variables, typically \( x \) or \( y \). Here, from the equation \( x = t - 3 \), it's straightforward to solve for \( t \) as \( t = x + 3 \).

Once you have solved for \( t \), the next move is to replace "\( t \)" in the other equation. By substituting into \( y = 3t - 7 \), we find \( y = 3(x + 3) - 7 \). By simplifying, this equation becomes \( y = 3x + 2 \). By this method, you've transformed the parametric form into a simple linear equation, which describes a straight line in the Cartesian coordinate plane without any parameters.

Graphing parametric curves might seem complex at first because you deal with equations involving a parameter. However, once you eliminate the parameter, you simplify the task significantly. The equation you derived, \( y = 3x + 2 \), represents a straight line.

To learn how to graph the curve properly, it's important to consider the domain of \( t \), which acts like a time variable indicating how the curve is traced out. For the exercise, \( t \) ranges from 0 to 3. This means you want to identify what portions of the line are actually drawn on the graph.

Calculate the corresponding \( x \) values when \( t \) is at its endpoints:

• When \( t = 0 \): \( x = t - 3 = 0 - 3 = -3 \)
• When \( t = 3 \): \( x = t - 3 = 3 - 3 = 0 \)

Mark these points on the graph as

• \( (-3, -7) \)
• \( (0, 2) \)

Draw a line between these points. The direction of increasing \( t \) is from \( (-3, -7) \) to \( (0, 2) \), indicating how the line is "traveled" as \( t \) goes from 0 to 3.

Linear equations are fundamental to mathematics and describe a straight line when graphed. The exercise demonstrates this through the equation \( y = 3x + 2 \). Understanding such equations is crucial as they form the basis of linear algebra and calculus.

A linear equation in two variables generally takes the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the \( y \)-intercept.

• In \( y = 3x + 2 \), the slope \( m \) is 3. This indicates how steep or flat the line is. For every unit increase in \( x \), \( y \) increases by 3 units.
• The \( y \)-intercept \( c \) is 2. This tells us where the line crosses the \( y \)-axis.

Linear equations provide a simple way to describe direct relationships between two variables. By finding the slope and intercept, you can quickly sketch their graph. In this exercise, the coherent transition from parametric equations to a linear one highlights how different mathematical expressions can represent the same geometric entity.